On $p$-compact group topologies on direct sums of ${\mathbb Q}$

TL;DR

The paper constructs large, torsion-free, countably compact topological groups based on direct sums of rationals, which are $p$-compact without non-trivial convergent sequences, under certain ultrafilter conditions.

Contribution

It provides the first examples of large, torsion-free, countably compact groups lacking non-trivial convergent sequences, using $p$-compact topologies and selective ultrafilters.

Findings

Existence of $p$-compact group topologies on ${f Q}^{(oldsymbol{rown})}$

Construction of arbitrarily large countably compact torsion-free groups

Groups lack non-trivial convergent sequences under specified conditions

Abstract

We prove that if $p$ is a selective ultrafilter then ${\mathbb Q}^{(\kappa)}$ has a $p$-compact group topology without non-trivial convergent sequences, for each infinite cardinal $\kappa =\kappa^\omega$. In particular, this gives the first arbitrarily large examples of countably compact groups without non-trivial convergent sequences that are torsion-free.

Equations

$$\bigcup _{n\in \omega }\mathrm{supp}{f}_{\xi }(n)\subseteq \xi \text{, for each }\xi \in [\omega ,\kappa ).$$

$$K_n.\varphi (\mu )\le \psi (\mu )\le \varrho (\mu )$$

$$\sum _{h\in \mathcal{C}}{\mathcal{M}}_{f,h}.W_h\subseteq U_f.$$

$$\overline{\sum _{\mu \in \mathrm{supp}{d}_0}{d}_0(\mu ).{\varrho }^0(\mu )}\cap \overline{\sum _{\mu \in \mathrm{supp}{d}_1}{d}_1(\mu ).{\varrho }^0(\mu )}=\varnothing ,$$

Full text

The first listed author has received financial support from FAPESP 2017/15709-6.

The second listed author has received financial support from FAPESP 2017/15502-2.

The third listed author has received financial support from FAPESP 2016/26216-8.

1. Introduction

1.1. Some history

Halmos proved that ${\mathbb{R}}$ can be endowed with a compact group topology, which in particular contains non-trivial convergent sequences [4]. Tkachenko and Yashenko showed from Martin’s Axiom that ${\mathbb{R}}$ can be endowed with a countably compact group topology without non-trivial convergent sequences [6]. Madariaga-Garcia and Tomita assumed the existence of $2^{\mathfrak{c}}$ selective ultrafilters to obtain a countably compact group topology on the free Abelian group of cardinality $2^{\mathfrak{c}}$ without non-trivial convergent sequences. The construction does not yield any example of larger cardinality [5].

Fuchs showed that there are no compact free Abelian groups [3] and Tomita showed that there are no free Abelian groups whose countable power is countably compact [7]. In particular, there are no $p$-compact free Abelian groups. Using what was called integer stacks, Tomita showed that there exists a group topology in the free Abelian group with ${\mathfrak{c}}$ generators whose every finite power is countably compact [8].

Assuming the existence of a selective ultrafilter $p$ and a cardinal arithmetic assumption weaker than $GCH$, Castro-Pereira and Tomita classified the torsion groups that admit a countably compact group topology without non-trivial convergent sequences [2]. These torsion Abelian groups coincide with the torsion Abelian groups that admit a $p$-compact group topology without non-trivial convergent sequences. In particular, it is shown that there exist large countably compact torsion groups without non-trivial convergent sequences.

Comfort called the direct sums of ${\mathbb{Q}}$ the test space for pseudocompactness for non-torsion groups. Our aim is to show the result in the abstract. This gives the first example of arbitrarily large torsion free countably compact groups without non-trivial convergent sequences. Also, it follows that ${\mathbb{R}}$ can be endowed with a $p$-compact group topology without non-trivial convergent sequences whenever $p$ is a selective ultrafilter.

For the construction in this work, we will adapt the idea of integer stacks in [8], used for direct sums of ${\mathbb{Z}}$’s, to work with direct sums of ${\mathbb{Q}}$’s. There are some key changes in how the stack is organized and we will give a complete description of rational stacks without assuming knowledge of the previous construction.

We will also use some ideas in [9] to solve arc equations.

We notice that this construction uses the fact that the ultrapower of direct sums of ${\mathbb{Q}}$ is again a ${\mathbb{Q}}$-vector space. The ultrapower of a free Abelian group is no longer a free Abelian group and as mentioned above, there are no $p$-compact free Abelian groups.

1.2. Some basic notation

Fix an infinite cardinal $\kappa$ such that $\kappa=\kappa^{\omega}$.

Let $\mathbb{T}$ be the Abelian group $\mathbb{R}/\mathbb{Z}$.

Let $G$ be the Abelian additive group $\mathbb{Q}^{(\kappa)}=\{g\in\mathbb{Q}^{\kappa}:|\operatorname{\text{supp }}g|<\omega\}$ and let $H$ be the Abelian additive group $\mathbb{Z}^{(\kappa)}=\{g\in\mathbb{Z}^{\kappa}:|\operatorname{\text{supp }}g|<\omega\}$. If $C\subseteq\kappa$, let $\mathbb{Q}^{(C)}=\{g\in G:\operatorname{\text{supp }}g\subseteq C\}$.

Definition 1.1**.**

Given a $\mu\in\kappa$, we denote by $\chi_{\mu}$ the element of $G$ such that $\operatorname{\text{supp }}\chi_{\mu}=\{\mu\}$ and $\chi_{\mu}(\mu)=1$.

Given $\mu\in\kappa$, we define $\vec{\mu}$ the sequence such that $\vec{\mu}(n)=\mu$ for every $n\in\omega$.

If $A\subseteq\omega$ and $\zeta:\,A\longrightarrow\kappa$ then we define $\chi_{\zeta}\in G^{A}$ such that $\chi_{\zeta}(n)=\chi_{\zeta(n)}$ for each $n\in\text{dom }\zeta$.

Definition 1.2**.**

Given $f\in G^{A}$ and $s=(s_{n}:n\in A)$ a sequence of rational numbers, we will we denote by $s.f$ or $sf$ the function from $A$ into $G$ such that $(s.f)(n)=s_{n}.f(n)$, for each $n\in A$.

Definition 1.3**.**

Given ${\mathcal{A}}\subseteq G^{A}$ and $s=(s_{n}:n\in A)$ a sequence of rational numberss, we define $s\mathcal{A}=\{sf:f\in\mathcal{A}\}$. If $s:A\rightarrow\mathbb{Q}\setminus\{0\}$, we define $\frac{\mathcal{A}}{s}=\{\frac{1}{s}f:f\in\mathcal{A}\}$.

Given an ultrafilter $q$, we define a equivalence relation on $G^{\omega}$ by letting $f\simeq_{q}g$ iff $\{n\in\omega:f(n)=g(n)\}\in q$. We let $[f]_{q}$ be the equivalence class determined by $f$ and $G^{\omega}/q$ be $G^{\omega}/\simeq_{q}$. Notice that this set has a natural $\mathbb{Q}$-vector space structure.

A rough idea of the construction framework is the following:

• •

Define an associated family of stacks (Lemma 3.1).

• •

Solve arc equations in a level of a stack (Lemma 3.2).

• •

Find a relation between the arc equations related to a finite sequence of functions and some arc equations in the associated stack (Lemma 3.3).

• •

Find the levels of the stacks within an element of the ultrafilter $p$. Solve the arc equations in the stack and build the homomorphism. (Proof of Lemma 2.3 in Section 4).

2. Homomorphisms, arc functions and arc equations

The way to construct the group topology is through homomorphisms that preserve $p$-compactness for the representatives of a base for the ultrapower $p$. The homomorphisms are constructed by defining smaller and smaller arcs that will give an approximation of the homomorphism. These approximations are arc equations that need to have a solution inside an element of the ultrafilter.

Definition 2.1**.**

Let ${\mathbb{B}}=\{I+\mathbb{Z}:I\subset\mathbb{R}$ is an open interval$\}$ be the family of all the open arcs in ${\mathbb{T}}$, including ${\mathbb{T}}$ itself.

We will call an arc function a function $\phi:\,\kappa\longrightarrow{\mathbb{B}}$ such that $\{\xi\in\kappa:\,\phi(\xi)\neq{\mathbb{T}}\}$ is finite. This set will be called the support of the arc function. If the arcs $\phi(\xi)$ have length $\epsilon$ for each $\xi$ in the support of $\phi$, we will call it an $\epsilon$-arc function.

Given two arc functions $\psi$ and $\phi$, we will say that $\psi\leq\phi$ if $\psi(\xi)=\phi(\xi)$ or $\overline{\psi(\xi)}\subseteq\phi(\xi)$, for each $\xi\in\kappa$.

Given an arc function $\phi$ and a positive integer $S$, $S.\phi$ is the arc function with support $\operatorname{\text{supp }}\phi$ such that ($S.\phi)(\mu)=S.\phi(\mu)$ for every $\mu\in\kappa$.

Below we give the definition of what we will call solutions of an arc equation.

Definition 2.2**.**

An arc equation is a quintuple $(\phi,A,\mathcal{A},S,U)$ where $\phi$ is an arc function, $A\subseteq\omega$, ${\mathcal{A}}\subseteq({\mathbb{Z}}^{(\kappa)})^{A}$, $S$ is a positive integer and $U=(U_{f}:f\in\mathcal{A})$ is a family of elements of ${\mathbb{B}}$.

Given $n\in A$, an $n$-solution for the arc equation $(\phi,A,\mathcal{A},S,U)$ is an arc function $\psi$ with $S\psi\leq\phi$ such that $\sum_{\mu\in\operatorname{\text{supp }}f(n)}f(n)(\mu)\psi(\mu)\subseteq U_{f}$, for each $f\in{\mathcal{A}}$.

We will use arc functions as approximations of a homomorphism function in a countable subgroup. The solutions of these equations will depend on the notion of stacks that will be defined later in this work.

The goal is to prove the following:

Lemma 2.3** (Main Lemma).**

Fix a selective ultrafilter $p$. Let $\mathcal{F}\subseteq G^{\omega}$ be a countable collection of distinct elements mod $p$ such that $\{[f]_{p}:f\in\mathcal{F}\}\dot{\cup}\{[\chi_{\vec{\mu}}]_{p}:\,\mu\in\kappa\}$ is $\mathbb{Q}$-linearly independent in $G^{\omega}/p$.

Let $d,d_{0},d_{1}\in G\setminus\{0\}$ with $\operatorname{\text{supp }}d$, $\operatorname{\text{supp }}d_{0}$, $\operatorname{\text{supp }}d_{1}$ pairwise disjoint, and $C$ be a countably infinite subset of $\kappa$ such that $\omega\cup\operatorname{\text{supp }}d\cup\operatorname{\text{supp }}d_{0}\cup\operatorname{\text{supp }}d_{1}\cup\bigcup_{f\in\mathcal{F},n\in\omega}\operatorname{\text{supp }}f(n)\subseteq C$. For each $f\in\mathcal{F}$, choose $\xi_{f}\in C$.

Then there exists a homomorphism $\phi:\,{\mathbb{Q}}^{(C)}\longrightarrow{\mathbb{T}}$ such that

1. a)

$\phi(d)\neq 0$, $\phi(d_{0})\neq\phi(d_{1})$, and 2. b)

$p$-$\lim(\phi(\frac{1}{N}.f))=\phi(\frac{1}{N}.\chi_{\xi_{f}})$, for each $f\in\mathcal{F}$ and $N\in\omega$.

For the remaining of this section, let $\{f_{\alpha}:\,\omega\leq\alpha<\kappa\}$ be an enumeration of $G^{\omega}$ such that

[TABLE]

By applying the Main Lemma, we get the following result:

Lemma 2.4**.**

Fix a selective ultrafilter $p$. Let $I\subseteq[\omega,\kappa)$ be such that $\{[f_{\xi}]_{p}:\,\xi\in I\}\cup\{[\chi_{\vec{\mu}}]_{p}:\,\mu\in\kappa\}$ is a $\mathbb{Q}$-basis for $G^{\omega}/p$.

Let $d\in G\setminus\{0\}$, $D\in[I]^{<\omega}\setminus\{\varnothing\}$, $r\in G$ of support $D$ and $B\in p$. Let $C$ be a countably infinite subset of $\kappa$ such that $\omega\cup D\cup\operatorname{\text{supp }}d\subseteq C$ and $\bigcup_{n\in\omega}\operatorname{\text{supp }}f_{\xi}(n)\subseteq C$ for every $\xi\in C\cap I$.

Then there exists a homomorphism $\phi:\,{\mathbb{Q}}^{(C)}\longrightarrow{\mathbb{T}}$ such that

1. a)

$\phi(d)\neq 0$, 2. b)

$p$-$\lim(\phi(\frac{1}{N}.f_{\xi}))=\phi(\frac{1}{N}.\chi_{\xi})$, for each $\xi\in I\cap C$ and $N\in\omega$, and 3. c)

$(\phi(\sum_{\mu\in D}r(\mu)f_{\mu}(n)):n\in B)$ does not converge.

Proof.

Let $B^{\prime}\in p$ a subset of $B$ such that $(\sum_{\mu\in D}r(\mu)f_{\mu}(n):n\in B^{\prime})$ is a 1-1 sequence. Let ${\mathbb{A}}$ be an almost disjoint family on $B^{\prime}$ of cardinality ${\mathfrak{c}}$ and $h_{x}:\,\omega\longrightarrow\{\sum_{\mu\in D}r(\mu)f_{\mu}(n):\,n\in x\}$ be a bijection for each $x\in\mathbb{A}$.

Claim: There exist $x_{0},x_{1}\in{\mathbb{A}}$ such that $\{[f_{\xi}]_{p}:\,\xi\in C\cap I\}\cup\{[\chi_{\vec{\mu}}]_{p}:\,\mu\in\kappa\}\cup\{[h_{x_{0}}]_{p},[h_{x_{1}}]_{p}\}$ is a linearly independent subset.

Proof of the claim: Given $x_{0},x_{1}\in\mathbb{A}$, notice that $h_{x_{1}}(n)\neq h_{x_{2}}(n)$ for all but a finite numbers of $n$’s, so $[h_{x_{1}}]_{p}\neq[h_{x_{2}}]_{p}$. Since $\mathbb{Q}$ is countable, it follows that $\langle[h_{x}]_{p}:x\in\mathbb{A}\rangle$ has cardinality $\mathfrak{c}$, so there is $J\subseteq\mathbb{A}$ such that $|J|=\mathfrak{c}$ and that $([h_{x}]_{p}:x\in J)$ is linearly independent. Now notice that $\langle f_{\xi}:\xi\in I\cap C\rangle\oplus\langle\chi_{\vec{\xi}}:\xi\in C\rangle$ is countable, so there exists $x_{0},x_{1}\in J$ such that $\{[f_{\xi}]_{p}:\,\xi\in C\cap I\}\cup\{[\chi_{\vec{\mu}}]_{p}:\,\mu\in C\}\cup\{[h_{x_{0}}]_{p},[h_{x_{1}}]_{p}\}$ is linearly independent. Since all the supports of these elements are contained in $C$, it is straightforward to see that $\{[f_{\xi}]_{p}:\,\xi\in C\cap I\}\cup\{[\chi_{\vec{\mu}}]_{p}:\,\mu\in\kappa\}\cup\{[h_{x_{0}}]_{p},[h_{x_{1}}]_{p}\}$ is a linearly independent. ∎

Let ${\mathcal{F}}=\{f_{\xi}:\,\xi\in C\cap I\}\cup\{h_{x_{0}},h_{x_{1}}\}$. Set $\xi_{f}=\mu$ if $f=f_{\mu}$ for some $\mu\in C\cap I$ and $\xi_{f}=m_{i}$ if $f=h_{x_{i}}$ for some $i<2$ where $m_{0}\neq m_{1}$ and $m_{0},m_{1}\in\omega\setminus\operatorname{\text{supp }}d$. Let $d_{0}=\chi_{m_{0}}$, $d_{1}=\chi_{m_{1}}$ and $\phi$ be as in Lemma 2.3.

Clearly conditions a) and b) of Lemma 2.4 are satisfied.

Furthermore, $(\phi(h_{x_{i}}(k)):k\in\omega)$ has $\phi(\chi_{m_{i}})$ as an accumulation point for $i<2$. Since these sequences are reorderings of a subsequence of $(\phi(\sum_{\mu\in D}r(\mu)f_{\mu}(n)):n\in B)$ and $\phi(\chi_{m_{0}})\neq\phi(\chi_{m_{1}})$, it follows that c) is satisfied. ∎

Finally, we may extend the above homomorphism to $G$:

Lemma 2.5**.**

Fix a selective ultrafilter $p$. Let $I\subseteq[\omega,\kappa)$ be such that $\{[f_{\xi}]_{p}:\,\xi\in I\}\cup\{[\chi_{\vec{\mu}}]_{p}:\,\mu\in\kappa\}$ is a $\mathbb{Q}$-basis for $G^{\omega}/p$.

Let $d\in G\setminus\{0\}$, $D\in[I]^{<\omega}\setminus\{\varnothing\}$, $r:D\to\mathbb{Q}\setminus\{0\}$ and $B\in p$.

Then there exists a homomorphism $\phi:\,G\longrightarrow{\mathbb{T}}$ such that

1. a)

$\phi(d)\neq 0$, 2. b)

$p$-$\lim\phi(\frac{1}{N}.f_{\xi})=\phi(\frac{1}{N}.\chi_{\xi})$, for each $\xi\in I$ and $N\in\omega$, and 3. c)

$(\phi(\sum_{\mu\in D}r(\mu)f_{\mu}(n)):n\in B)$ does not converge.

Proof.

Let $C$ be a countably infinite subset of $\kappa$ such that $\omega\cup\operatorname{\text{supp }}d\subseteq C$ and $\bigcup_{n\in\omega}\operatorname{\text{supp }}f_{\xi}(n)\subseteq C$ for every $\xi\in C\cap I$. Such a $C$ exists by standard closing off arguments. Let $(\xi_{\alpha}:\alpha<\kappa)$ be a strictly increasing enumeration of $\kappa\setminus C$. Let $\phi$ be as in the Main Lemma.

For each $\alpha<\kappa$, let $C_{\alpha}=C\cup\{\xi_{\beta}:\beta<\alpha\}$ (so $C_{0}=C$ and $C_{\kappa}=\kappa$). Notice that for each $\alpha$ and $n\in\omega$, $\operatorname{\text{supp }}f_{\xi_{\alpha}}(n)\subseteq\xi_{\alpha}\subseteq C_{\alpha}$.

Recursively we define homomorphisms $\phi_{\alpha}:\mathbb{Q}^{(C_{\alpha})}\rightarrow\mathbb{T}$ for $\alpha\leq\kappa$ satisfying:

1. a)

$\phi_{0}=\phi$, 2. b)

$\phi_{\beta}\subseteq\phi_{\alpha}$ whenever $\beta\leq\alpha\leq\kappa$, and 3. c)

$p$-$\lim\phi_{\alpha}(\frac{1}{N}.f_{\xi})=\phi(\frac{1}{N}.\chi_{\xi})$, for each $\xi\in I\cap C_{\alpha}$ and $N\in\omega$.

We let $\phi_{0}=\phi$. For the limit step, just take unions. For the successor step $\alpha+1$ we proceed as follows:

Notice that $\mathbb{Q}^{(C_{\alpha+1})}=\mathbb{Q}^{(C_{\alpha})}\oplus\{q\chi_{\xi_{\alpha}}:q\in\mathbb{Q}\}$.

First, we define $\tilde{\phi}_{\alpha}:\{q\chi_{\xi_{\alpha}}:q\in\mathbb{Q}\}\rightarrow\mathbb{T}$ by letting $\tilde{\phi}_{\alpha}(\frac{M}{N}\chi_{{\xi_{\alpha}}})=M(p$-$\lim\phi_{\xi_{\alpha}}(\frac{1}{N}.f_{\xi_{\alpha}}))$· Since multiplying a group element by an integer is a continuous function and since $\phi_{\alpha}$ is a homomorphism, it follows that $\tilde{\phi}_{\alpha}$ is well-defined and a group homomorphism. Now let $\phi_{\alpha+1}=\phi_{\alpha}\oplus\tilde{\phi}_{\alpha}$.

The required homomorphism is $\phi_{\kappa}$.

∎

From this lemma follows the result in the title:

Theorem 2.6**.**

Assume that $p$ is a selective ultrafilter and $\kappa=\kappa^{\omega}$ is an infinite cardinal. Then, there exists a $p$-compact group topology on $G={\mathbb{Q}}^{(\kappa)}$ without non-trivial convergent sequences.

Proof.

Let $p$ be a selective ultrafilter, let $I$ be as in the previous theorem. For each $d\in G\setminus\{0\}$,$D\in[I]^{<\omega}\setminus\{\varnothing\}$, $r:D\to\mathbb{Q}\setminus\{0\}$ and $B\in p$, let $\phi_{d,D,r,B}:G\rightarrow\mathbb{T}$ be as in the previous theorem.

The group topology induced by these homomorphisms is such that the $p$-$\lim(\frac{1}{N}.f_{\xi})=\frac{1}{N}.\chi_{\xi}$, for each $\xi\in I$ and $N\in\omega$.

If $h$ is any element of $G^{\omega}$, there exist families $(r_{\xi}:\xi\in I)$ and $(s_{\mu}:\mu\in\kappa)$ of rational numbers where all but a finite quantity of them is [math] such that:

$[h]_{p}=\sum_{\xi\in I}r_{\xi}[f_{\xi}]_{p}+\sum_{\mu\in\kappa}s_{\mu}[\chi_{\vec{\mu}}]_{p}$. Then $\sum_{\xi\in I}r_{\xi}.\chi_{\xi}+\sum_{\mu\in\kappa}s_{\mu}.\chi_{\mu}$ is the $p$-limit of $h$. Therefore, $G$ is $p$-compact.

To check that there are no non-trivial convergent sequences, fix a 1-1 sequence $g$. Let $(r_{\xi}:\xi\in I)$ and $(s_{\mu}:\mu\in\kappa)$ be families of rational numbers where all but a finite quantity of them is [math] such that:

$[g]_{p}=\sum_{\xi\in I}r_{\xi}[f_{\xi}]_{p}+\sum_{\mu\in\kappa}s_{\mu}[\chi_{\vec{\mu}}]_{p}$. Then there are $B\in p$ and $D\in[I]^{<\omega}\setminus\{\varnothing\}$ such that: $g(n)=\sum_{\xi\in D}r_{\xi}f_{\xi}(n)+\sum_{\mu\in\kappa}s_{\mu}\chi_{\mu}$, for all $n\in B$. By Lemma 2.5(c), we have that $(\phi_{d,D,r,B}(\sum_{\xi\in D}r_{\xi}f_{\xi}(n)):n\in B)$ does not converge in $\mathbb{T}$, and so $(\sum_{\xi\in D}r_{\xi}f_{\xi}(n):n\in B)$ does not converge in $G$. Since $\sum_{\mu\in\kappa}s_{\mu}\chi_{\mu}$ is constant, it follows that $(g(n):n\in B)$ does not converge,and therefore $g$ does not converge. ∎

3. A preliminar discussion on rational stacks

We will start this section with an informal discussion about rational stacks. A rational stack will be a nonuple $\langle\mathcal{B},\nu,\zeta,K,A,k_{0},k_{1},l,T\rangle$, where:

• •

$A\subseteq\omega$ is infinite,

• •

$k_{0}\leq k_{1}$ are natural numbers with $k_{1}>0$,

• •

$l:k_{1}\rightarrow\omega$,

• •

$\nu:k_{0}\rightarrow\kappa$,

• •

$\zeta:k_{1}\rightarrow\kappa^{\omega}$,

• •

$K:A\rightarrow\omega\setminus 2$ is such that for every $n\in A$, $n!T\mid K_{n}$,

• •

$\mathcal{B}=({{\mathcal{B}}}_{i,j}:\,i<k_{1},j<l_{i})$ is such that each $\mathcal{B}_{i,j}\subseteq H^{\omega}$ is finite.

• •

$T>0$ is an integer.

In order to be a rational stack, this nonuple must satisfy additional properties. The full definition of rational stack will be given in Section 5. This definition was designed to solve arc equations to construct the homomorphisms we want.

Roughly speaking, we associate each finite family of sequences to a stack. We transform the arc equations associated to this finite family to arc equations associated to the stack, solve the arc equation using the properties of the stack then return to a solution of the original arc equations. We want to solve infinitely many equations, thus, this process is made back and forth, including more equations at each step. At each stage, the stack is different and there is no containment relation between the stacks, even though we use a larger finite subfamily of sequences.

The idea is that before each iteration we have arc equations related to a certain arc size. We might need to shrink the arcs when we write the arc equations related to the stack so that the solution of arc equation associated to the stack can be transformed into a solution of the original equations. After each step, the output arcs will be smaller and will be the input arcs for the next iteration. We have to estimate the size of the output arcs according to its input arc so that the sets of arc equations can be solved ultrafilter often. For this, we use the happiness of the selective ultrafilter.

Before we even define the stack, we will list the main results that motivated its definition.

The Lemma below associates each finite subset of functions to a stack that will be used to solve arc equations associated to this family.

Lemma 3.1**.**

Let $B\in p$ and $\mathcal{G}$ be a finite subset of $G^{\omega}$ whose elements are distinct mod $p$ and none of them is constant mod $p$ such that $\{[f]_{p}:f\in\mathcal{G}\}\cup\{[\chi_{\vec{\nu}}]_{p}:\nu\in\kappa\}$ is linearly independent. Then there exists a rational stack $\mathcal{S}=\langle\mathcal{B},\nu,\zeta,K,A,k_{0},k_{1},l,T\rangle$ such that, by defining ${\mathcal{A}}={\mathcal{G}}\cup\{\chi_{\vec{\nu_{i}}}:\,i<k_{0}\}$ and ${\mathcal{C}}=\frac{\bigcup_{i<k_{1},j<l_{i}}{\mathcal{B}}_{i,j}}{{K}}$, there exist ${\mathcal{M}}:\mathcal{A}\times\mathcal{C}\rightarrow\mathbb{Z}$, ${\mathcal{N}}:\mathcal{C}\times\mathcal{A}\rightarrow\mathbb{Z}$ satisfying:

1. (1)

$\{[f]_{p}:\,f\in{\mathcal{A}}\}$ and $\{[h]_{p}:\,h\in{\mathcal{C}}\}$ generate the same subspace; 2. (2)

$f(n)=\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}h(n)$, for each $n\in A$ and $f\in{\mathcal{A}}$, 3. (3)

$h(n)=\frac{1}{T^{2}}.\sum_{f\in{\mathcal{A}}}{\mathcal{N}}_{h,f}f(n)$, for each $n\in A$ and $h\in{\mathcal{C}}$, 4. (4)

$K\mathcal{A}\subseteq H^{\omega}$, 5. (5)

$K\mathcal{C}\subseteq H^{\omega}$, and 6. (6)

$A\subseteq B$.

Proof.

The proof is quite technical and will be presented in a later section. ∎

Notice that if we interpret $\mathcal{M}$ and $\mathcal{N}$ as matrices, then $\mathcal{M}$ and $\frac{1}{T^{2}}\mathcal{N}$ are inverse matrices.

Lemma 3.2**.**

Let $\mathcal{S}$, ${\mathcal{A}}$, ${\mathcal{C}}$, ${\mathcal{M}}$ and ${\mathcal{N}}$ be as in Lemma 3.1. Let $\epsilon$ be a positive real and $D$ be a finite subset of $\kappa$. Then there exist $B\subseteq A$ cofinite in $A$ and a family of positive real numbers $(\gamma_{n}:n\in B)$ such that:

For every $n\in B$, for every family $(W_{h}:\,h\in{\mathcal{C}})$ of open arcs of length $\epsilon$, and for every arc function $\psi$ of length $\epsilon$ such that $\operatorname{\text{supp }}\psi\subseteq D\setminus\{\nu_{0},\ldots\nu_{k_{0}-1}\}$, there exists an $n$-solution of length $\gamma_{n}$ for the arc equation $(\psi,B,K.\mathcal{C},K_{n},W)$.

Proof.

The proof is quite technical and will be presented in a later section. ∎

Lemma 3.3**.**

Let $\mathcal{S}$, ${\mathcal{A}}$, ${\mathcal{C}}$, ${\mathcal{M}}$ and ${\mathcal{N}}$ be as in Lemma 3.1. Let $\delta$ be a positive real such that $\epsilon=\frac{\delta}{\sum_{f\in{\mathcal{A}},h\in{\mathcal{C}}}|{\mathcal{M}}_{f,h}|}<1$.

Let $(U_{f}:\,f\in{\mathcal{A}})$ be a family of open arcs of length $\delta$. Let $\varrho$ be an arc function of length $\delta$ such that $U_{\chi_{\vec{\nu}_{i}}}=\varrho(\nu_{i})$ for $i<k_{0}$. Furthermore, assume that $\{\nu_{i}:\,i<k_{0}\}\subseteq\operatorname{\text{supp }}\varrho$.

Then there exist $(W_{h}:\,h\in{\mathcal{C}})$ a family of open arcs of length $\epsilon$ and $\psi$ an $\epsilon$-arc function with support $\operatorname{\text{supp }}\varrho\setminus\{\nu_{0},\ldots,\nu_{k_{0}-1}\}$ such that for every $n\in A$, every $n$-solution for the arc equation $(\psi,A,K.\mathcal{C},K_{n},W)$ is an $n$-solution for $(\varrho,A,K.\mathcal{A},K_{n},U)$.

Proof.

Given $f\in\mathcal{A}$, let $y_{f}\in{\mathbb{R}}$ be such that $y_{f}+\mathbb{Z}$ is the center of the arc $U_{f}$.

For each $h\in\mathcal{C}$, let $z_{h}=\sum_{f\in{\mathcal{A}}}{\mathcal{N}}_{h,f}.\frac{y_{f}}{T^{2}}$. Since ${\mathcal{N}}$ is an integer matrix, it follows that $z_{h}+{\mathbb{Z}}=\sum_{f\in{\mathcal{A}}}{\mathcal{N}}_{h,f}.(\frac{y_{f}}{T^{2}}+{\mathbb{Z}})$. Let $W_{h}$ be an arc centered in $z_{h}+{\mathbb{Z}}$ whose length is $\epsilon$.

Let $\psi(\mu)$ be an arc of same center as $\varrho(\mu)$ of length $\epsilon$ for each $\mu\in\operatorname{\text{supp }}\varrho\setminus\{\nu_{0},\ldots,\nu_{k_{0}-1}\}$ and $\psi(\nu_{i})=\mathbb{T}$ for each $i<k_{0}$.

Suppose $\phi$ is an $n$-solution for $(\psi,A,K.\mathcal{C},K_{n},W)$. Then $\sum_{\mu\in\operatorname{\text{supp }}h(n)}K_{n}.h(n)(\mu).\phi(\mu)\subseteq W_{h}$, for each $h\in{\mathcal{C}}$. Also, we have that, for each $\mu\in\kappa\setminus\{\nu_{0},\ldots,\nu_{k_{0}-1}\}$:

[TABLE]

Let $f\in\mathcal{A}$.

We have that for each $\mu$, $\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}.K_{n}.h(n)(\mu).\phi(\mu)=K_{n}.f(n)(\mu)\phi(\mu)$. Therefore, $\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}.\sum_{\mu\in\operatorname{\text{supp }}h(n)}K_{n}.h(n)(\mu).\phi(\mu)=\sum_{\mu\in\operatorname{\text{supp }}f(n)}K_{n}.f(n)(\mu).\phi(\mu)$.

It then follows that $\sum_{\mu\in\operatorname{\text{supp }}f(n)}K_{n}.f(n)(\mu).\phi(\mu)\subseteq\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}.W_{h}$. The arc $\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}.W_{h}$ is centered in $\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}.(z_{h}+\mathbb{Z})=\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}\sum_{g\in{\mathcal{A}}}{\mathcal{N}}_{h,g}.(\frac{y_{g}}{T^{2}}+{\mathbb{Z}})=\sum_{g\in{\mathcal{A}}}\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}{\mathcal{N}}_{h,g}.(\frac{y_{g}}{T^{2}}+{\mathbb{Z}})=y_{f}+{\mathbb{Z}}$, and has length $\epsilon\cdot\sum_{h\in{\mathcal{C}}}|{\mathcal{M}}_{f,h}|\leq\delta$. Therefore,

[TABLE]

Thus, $\phi$ is an $n$-solution for $(\varrho,A,K.\mathcal{A},K_{n},U)$, as required, provided that we show $K_{n}.\phi\leq\varrho$.

From $\eqref{permaps}$, if $f=\chi_{\vec{\nu_{i}}}$ then $K_{n}.\phi(\nu_{i})=K_{n}\chi_{\vec{\nu_{i}}}(n)(\nu_{i})\phi(\nu_{i})\subseteq U_{f}=\varrho(\nu_{i})$, hence $K_{n}.\phi(\nu_{i})\leq\varrho(\nu_{i})$ for each $0\leq i<k_{0}$. This and (# ‣ 3) implies that $K_{n}\phi\leq\varrho$.

∎

4. Selective ultrafilters and the proof of the Main Lemma using the properties of the stacks

Our goal in this section is to prove Lemma 2.3 using the Lemmas in the previous section. First we state the following Lemma:

Lemma 4.1**.**

Fix a selective ultrafilter $p$. Let $\mathcal{F}\subseteq G^{\omega}$ be a countable collection of distinct elements mod $p$ such that $\{[f]_{p}:f\in\mathcal{F}\}\dot{\cup}\{[\chi_{\vec{\mu}}]_{p}:\,\mu\in\kappa\}$ is $\mathbb{Q}$-linearly independent in $G^{\omega}/p$.

Let $d,d_{0},d_{1}\in G\setminus\{0\}$ with $\operatorname{\text{supp }}d$, $\operatorname{\text{supp }}d_{0}$, $\operatorname{\text{supp }}d_{1}$ pairwise disjoint, and $C$ be a countably infinite subset of $\kappa$ such that $\omega\cup\operatorname{\text{supp }}d\cup\operatorname{\text{supp }}d_{0}\cup\operatorname{\text{supp }}d_{1}\cup\bigcup_{f\in\mathcal{F},n\in\omega}\operatorname{\text{supp }}f(n)\subseteq C$. For each $f\in\mathcal{F}$, choose $\xi_{f}\in C$. Let $(\mathcal{F}^{n}:n\in\omega)$ be an increasing sequence of finite sets whose union is $\mathcal{F}$.

Then there exist stacks $\mathcal{S}^{m}=\langle\mathcal{B}^{m},\nu^{m},\zeta^{m},K^{m},A^{m},k_{0}^{m},k_{1}^{m},l^{m},T^{m}\rangle$ and ${\mathcal{A}}^{m}$, ${\mathcal{C}}^{m}$, $\mathcal{M}^{m}$, $\mathcal{N}^{m}$ related to the stacks as in Lemma 3.1, $r:\omega\cup\{-1\}\rightarrow\omega$ such that $r[\omega]\in p$ and $r(-1)=0$, and a sequence of arc functions $(\varrho^{r(m)}:m\geq-1)$ with $C\subseteq\bigcup_{m\geq-1}\operatorname{\text{supp }}\varrho^{r(m)}$, satisfying the following:

1. a)

$0\notin\overline{\sum_{\mu\in\operatorname{\text{supp }}d}d(\mu).\varrho^{0}(\mu)}$ and $\overline{\sum_{\mu\in\operatorname{\text{supp }}d_{0}}d_{0}(\mu).\varrho^{0}(\mu)}\cap\overline{\sum_{\mu\in\operatorname{\text{supp }}d_{1}}d_{1}(\mu).\varrho^{0}(\mu)}=\varnothing$. 2. b)

For every $m^{\prime},m\geq-1$ with $m^{\prime}\leq m$ and for every $\xi\in\operatorname{\text{supp }}\varrho^{r(m+1)}$, we have $\left(\prod_{i=m^{\prime}}^{m}K^{r(i)}_{r(i+1)}\right).\varrho^{r(m+1)}(\xi)\subseteq\varrho^{r(m^{\prime})}(\xi)$ and $\left(\prod_{i=m^{\prime}}^{m}K^{r(i)}_{r(i+1)}\right).\varrho^{r(m+1)}(\xi)$ has length $\leq\frac{1}{2^{r(m)+1}}$. 3. c)

For every $m\geq-1$ and $f\in\mathcal{F}^{r(m)}$, we have $\sum_{\mu\in\operatorname{\text{supp }}f(r(m+1))}\ f(r(m+1))(\mu)K^{r(m)}_{r(m+1)}\varrho^{r(m+1)}(\mu)\subseteq\varrho^{r(m)}(\xi_{f})$. 4. d)

For every $m,m^{\prime}\geq-1$ with $m^{\prime}\leq m$ and $f\in\mathcal{F}^{r(m)}$, we have $\sum_{\mu\in\operatorname{\text{supp }}f(r(m+1))}\ f(r(m+1))(\mu)\left(\prod_{i=m^{\prime}}^{m}K^{r(i)}_{r(i+1)}\right)\varrho^{r(m+1)}(\mu)\subseteq\varrho^{r(m^{\prime})}(\xi_{f})$. 5. e)

For every $m\geq-1$, $\operatorname{\text{supp }}\varrho^{r(m)}\subseteq\operatorname{\text{supp }}\varrho^{r(m+1)}$.

Proof.

Let $p$, $\mathcal{F}$, $d$, $d_{0}$, $d_{1}$,$C$ and $(\xi_{f}:f\in\mathcal{F})$ be given. Write $C$ as an increasing sequence of finite sets $(C^{n}:n\in\omega)$ such that for each $n\in\omega$, $\bigcup\{\operatorname{\text{supp }}f(k):f\in\mathcal{F}^{n}\text{ and }k\leq n\}\subset C^{n}$, and $\operatorname{\text{supp }}d\cup\operatorname{\text{supp }}d_{0}\cup\operatorname{\text{supp }}d_{1}\subset C^{0}$.

Apply Lemma 3.1 to $\mathcal{G}=\mathcal{F}^{0}$ and $B=\omega$ to obtain a rational stack $\mathcal{S}^{0}=\langle\mathcal{B}^{0},\nu^{0},\zeta^{0},K^{0},A^{0},k_{0}^{0},k_{1}^{0},l^{0},T^{0}\rangle$ and ${\mathcal{A}}^{0}$, ${\mathcal{C}}^{0}$, ${\mathcal{M}}^{0}$ and ${\mathcal{N}}^{0}$ satisfying (1)-(7) as in the Lemma.

Fix $\delta^{0}\in\mathbb{R}$ such that $0<\delta^{0}<1$ and $\varrho^{0}$ a $\delta^{0}$-arc function such that $0\notin\overline{\sum_{\mu\in\operatorname{\text{supp }}d}d(\mu).\varrho^{0}(\mu)}$ and $\overline{\sum_{\mu\in\operatorname{\text{supp }}d_{0}}d_{0}(\mu).\varrho^{0}(\mu)}\cap\overline{\sum_{\mu\in\operatorname{\text{supp }}d_{1}}d_{1}(\mu).\varrho^{0}(\mu)}=\varnothing$. We will also assume that $C^{0}\cup\{\nu_{i}^{0}:i<k^{0}_{0}\}\subseteq\operatorname{\text{supp }}\varrho^{0}$.

Let $\epsilon^{0}=\frac{\delta^{0}}{\sum_{f\in{\mathcal{A}^{0}},h\in{\mathcal{B}^{0}}}|{\mathcal{M}}^{0}_{f,h}|}$. Notice that with this $\epsilon^{0}$ we may apply Lemma 3.3.

Now we apply Lemma 3.2 with $D=C_{0}$ to obtain $B^{0}\subseteq A^{0}\setminus 1$ and $\boldsymbol{\gamma}^{0}=(\gamma^{0}_{n}:n\in B^{0})$ as in the Lemma.

Suppose that $B^{t}\in p$ with $(B^{t}:\,t\leq m)$ decreasing family of subsets of $\omega$, $\gamma^{t}_{n}$ for $t\leq m$ and $n\in B^{t}$ are defined.

Define $\delta^{m+1}=\frac{1}{2^{m+2}}.\frac{1}{\prod_{i,n\leq m+1}K^{i}_{n}}.\min(\{\gamma_{n}^{t}:\,t<n\leq m+2,n\in B^{t}\}\cup\{1\})$.

Apply Lemma 3.1 with $\mathcal{G}=\mathcal{F}^{m+1}$ and $B=B^{m}$ to obtain a stack $\mathcal{S}^{m+1}=\langle\mathcal{B}^{m+1},\nu^{m+1},\zeta^{m+1},K^{m+1},A^{m+1},k_{0}^{m+1},k_{1}^{m+1},l^{m+1},T^{m+1}\rangle$ and ${\mathcal{A}}^{m+1}$, ${\mathcal{C}}^{m+1},\mathcal{M}^{m+1},\mathcal{N}^{m+1}$ related to the stack as in the lemma. Then $A^{m+1}\subseteq B^{m}$. Let $\epsilon^{m+1}=\frac{\delta^{m+1}}{\sum_{f\in{\mathcal{A}}^{m+1},h\in{\mathcal{B}}^{m+1}}|{\mathcal{M}}^{m+1}_{f,h}|}$. Notice that with this $\epsilon^{m+1}$ we may apply Lemma 3.3.

Now we apply Lemma 3.2 with $D=C_{m+1}$ using to obtain $B^{m+1}\subseteq A^{m+1}\setminus m+2$ and $\boldsymbol{\gamma}^{m+1}=(\gamma^{m+1}_{n}:n\in B^{m+1})$ as in the Lemma.

We will use the happiness of the selective ultrafilter $p$: the sets constructed previously $B^{0}\supseteq B^{1}\ldots$ are all elements of $p$, so there exists a function $r\in\omega^{\omega}$ such that $r[\omega]\in p$, $r(0)\in B^{0}$ and, for all $n\in\omega$, $r(n+1)\in B^{r(n)}$.

Define $U^{0}=(U^{0}_{f}:f\in\mathcal{A}^{0})$, where $U^{0}_{f}=\varrho^{0}(\xi_{f})$ if $f\in\mathcal{F}^{0}$ or $U^{0}_{f}=\varrho^{0}(\nu^{0}_{i})$ if $f=\chi_{\vec{\nu}^{0}_{i}}$. By Lemma 3.3 applied on stage 0, we obtain $\psi$ and $W$ as in the conclusion of Lemma 3.3. Now, according to the conclusion of Lemma 3.2 used in stage 0 of the construction, since $r(0)\in B^{0}$, we obtain a $r(0)$-solution $\phi^{0}$ of length $\gamma^{0}_{r(0)}$ to the arc equation $(\psi,B^{0},K^{0}.\mathcal{C}^{0},K^{0}_{r(0)},W)$.

Now, using the conclusion of Lemma 3.3, we have that $\phi^{0}$ is a $r(0)$-solution to $(\varrho^{0},B^{0},K^{0}.\mathcal{A}^{0},K^{0}_{r(0)},U^{0})$.

In particular, we have, for each $f\in\mathcal{F}^{0}$,

$\sum_{\mu\in\operatorname{\text{supp }}f(r(0))}\ f(r(0))(\mu)K^{0}_{r(0)}\phi^{0}(\mu)\subseteq\varrho^{0}(\xi_{f})$.

Let $r(-1)=0$. We can inductively to construct $\phi^{r(m)}$, $\varrho^{r(m)}$ a $\delta^{r(m)}$-arc function with $C^{r(m)}\subseteq\operatorname{\text{supp }}\varrho^{r(m)}$, and $U^{r(m)}$ such that:

For every $m\geq-1$, $\phi^{r(m)}$ is a $r(m+1)$-solution of length $\gamma^{r(m)}_{r(m+1)}$ of the arc equation $(\varrho^{r(m)},B^{r(m)},K^{r(m)}.\mathcal{A}^{r(m)},K^{r(m)}_{r(m+1)},U^{r(m)})$, 2. 2)

For every $m\geq-1$, $\varrho^{r(m+1)}\leq\phi^{r(m)}$, 3. 3)

For every $m^{\prime},m\geq-1$ with $m^{\prime}\leq m$ and for every $\xi\in\operatorname{\text{supp }}\varrho^{r(m+1)}$, we have $\left(\prod_{i=m^{\prime}}^{m}K^{r(i)}_{r(i+1)}\right).\varrho^{r(m+1)}(\xi)\subseteq\varrho^{r(m^{\prime})}(\xi)$ and $\left(\prod_{i=m^{\prime}}^{m}K^{r(i)}_{r(i+1)}\right).\varrho^{r(m+1)}(\xi)$ has length $\leq\frac{1}{2^{r(m)+1}}$, 4. 4)

For every $m\geq-1$ and $f\in\mathcal{F}^{r(m)}$, we have $\sum_{\mu\in\operatorname{\text{supp }}f(r(m+1))}\ f(r(m+1))(\mu)K^{r(m)}_{r(m+1)}\varrho^{r(m+1)}(\mu)\subseteq\varrho^{r(m)}(\xi_{f})$, 5. 5)

For every $m,m^{\prime}\geq-1$ with $m^{\prime}\leq m$ and $f\in\mathcal{F}^{r(m)}$, we have $\sum_{\mu\in\operatorname{\text{supp }}f(r(m+1))}\ f(r(m+1))(\mu)\left(\prod_{i=m^{\prime}}^{m}K^{r(i)}_{r(i+1)}\right)\varrho^{r(m+1)}(\mu)\subseteq\varrho^{r(m^{\prime})}(\xi_{f})$, 6. 6)

For every $m\geq-1$, $\operatorname{\text{supp }}\varrho^{r(m)}\subseteq\operatorname{\text{supp }}\varrho^{r(m+1)}$, and 7. 7)

$U^{r(m)}=(U^{r(m)}_{f}:f\in\mathcal{A}^{r(m)})$, where $U^{r(m)}_{f}=\varrho^{r(m)}(\xi_{f})$ if $f\in\mathcal{F}^{r(m)}$ or $U^{r(m)}_{f}=\varrho^{r(m)}(\nu^{r(m)}_{i})$ if $f=\chi_{\vec{\nu}^{r(m)}_{i}}$.

The base of the recursion is already done. Suppose the construction is done until step $m$ and let us define $\varrho^{r(m+1)}$ and $\phi^{r(m+1)}$.

Let $\varrho^{r(m+1)}$ be a $\delta^{r(m+1)}$-arc funtion such that $\operatorname{\text{supp }}\varrho^{r(m)}\cup C^{r(m+1)}\cup\{\nu_{i}^{r(m+1)}:i<k^{r(m+1)}_{0}\}\subseteq\operatorname{\text{supp }}\varrho^{r(m+1)}$ and $\varrho^{r(m+1)}\leq\phi^{r(m)}$.

Now define $U^{r(m+1)}=(U^{r(m+1)}_{f}:f\in\mathcal{A}^{r(m+1)})$, where $U^{r(m+1)}_{f}=\varrho^{r(m+1)}(\xi_{f})$ if $f\in\mathcal{F}^{r(m+1)}$ or $U^{r(m+1)}_{f}=\varrho^{r(m+1)}(\nu^{r(m+1)}_{i})$ if $f=\chi_{\vec{\nu}^{r(m+1)}_{i}}$. By Lemma 3.3 applied on stage $m+1$, we obtain $\psi$ and $W$ as in the conclusion of Lemma 3.3. Now, according to the conclusion of Lemma 3.2 used in stage $m+1$ of the construction, since $r(m+2)\in B^{r(m+1)}$, we obtain a $r(m+2)$-solution $\phi^{r(m+1)}$ of length $\gamma^{r(m+1)}_{r(m+2)}$ to the arc equation $(\psi,B^{r(m+1)},K^{r(m+1)}.\mathcal{C}^{r(m+1)},K^{r(m+1)}_{r(m+2)},W)$.

Now, using the conclusion of Lemma 3.3, we have that $\phi^{r(m+1)}$ is a $r(m+2)$-solution to $(\varrho^{r(m+1)},B^{r(m+1)},K^{r(m+1)}.\mathcal{A}^{r(m+1)},K^{r(m+1)}_{r(m+2)},U^{r(m+1)})$.

With , $\varrho^{r(m+1)}$ and $\phi^{r(m+1)}$ thus defined, items 1), 2), 6) and 7) of the recursion are immediately satisfied.

In order to verify item 3): the second statement follows from the definition of $\delta^{r(m+1)}$. As for the first, use items 1) and 2) and then use item 3) iteratively.

Item 4) follows from items 1) and 2) and the definiton of $U^{r(m)}$.

Item 5) follows from multiplying the expression in 4) by $\left(\prod_{i=m^{\prime}}^{m-1}K^{r(i)}_{r(i+1)}\right)$ and then applying item 3) for $m^{\prime}$ and $m-1$.

Now that the recursion is complete, notice that items a)-e) of the statement of the Lemma are clearly satisfied.

∎

Lemma** (Main Lemma).**

Fix a selective ultrafilter $p$. Let $\mathcal{F}\subseteq G^{\omega}$ be a countable collection of distinct elements mod $p$ such that $\{[f]_{p}:f\in\mathcal{F}\}\dot{\cup}\{[\chi_{\vec{\mu}}]_{p}:\,\mu\in\kappa\}$ is $\mathbb{Q}$-linearly independent in $G^{\omega}/p$.

Let $d,d_{0},d_{1}\in G\setminus\{0\}$ with $\operatorname{\text{supp }}d$, $\operatorname{\text{supp }}d_{0}$, $\operatorname{\text{supp }}d_{1}$ pairwise disjoint, and $C$ be a countably infinite subset of $\kappa$ such that $\omega\cup\operatorname{\text{supp }}d\cup\operatorname{\text{supp }}d_{0}\cup\operatorname{\text{supp }}d_{1}\cup\bigcup_{f\in\mathcal{F},n\in\omega}\operatorname{\text{supp }}f(n)\subseteq C$. For each $f\in\mathcal{F}$, choose $\xi_{f}\in C$.

Then there exists a homomorphism $\phi:\,{\mathbb{Q}}^{(C)}\longrightarrow{\mathbb{T}}$ such that

1. a)

$\phi(d)\neq 0$, $\phi(d_{0})\neq\phi(d_{1})$, and 2. b)

$p$-$\lim(\phi(\frac{1}{P}.f))=\phi(\frac{1}{P}.\chi_{\xi_{f}})$, for each $f\in\mathcal{F}$ and $P\in\omega\setminus\{0\}$.

Proof.

Let $\mathcal{S}^{m}$, $\mathcal{A}^{m}$, $\mathcal{C}^{m}$, $\mathcal{M}^{m}$, $\mathcal{N}^{m}$, $\mathcal{F}^{m}$ and $\varrho^{r(m)}$, $(m\in\omega)$, and $\varrho^{0}$ be as in the previous lemma.

For each $m\in\omega$, let $Q_{m}=\prod_{i=-1}^{m-1}K^{r(i)}_{r(i+1)}$. Now given a positive integer $m^{\prime}$ and $\xi\in C\cap\operatorname{\text{supp }}\varrho^{r(m^{\prime})}$, define $\phi\left(\frac{1}{Q_{m^{\prime}}}.\chi_{\xi}\right)$ as the unique element of $\bigcap_{m\geq m^{\prime}}\frac{Q_{m}}{Q_{m^{\prime}}}\varrho^{r(m)}(\xi)$. Furthermore, if $P$ divides $Q_{m^{\prime}}$ then define $\phi(\frac{1}{P}.\chi_{\xi})=\frac{Q_{m^{\prime}}}{P}\phi\left(\frac{1}{Q_{m^{\prime}}}.\chi_{\xi}\right)$. Then, since $n!|K_{n}^{m}$ for every $n,m$, $\phi\left(\frac{1}{P}.\chi_{\xi}\right)$ is well-defined and does not depend on $Q_{m^{\prime}}$ and $\phi$ can be extended to a homomorphism.

Notice that since $0\notin\overline{\sum_{\mu\in\operatorname{\text{supp }}d}d(\mu).\varrho^{0}(\mu)}$ and

[TABLE]

it follows that $\phi(d)\neq 0$ and $\phi(d_{0})\neq\phi(d_{1})$.

Let $f\in\mathcal{F}$ and $P$ be a positive integer. Let $M$ be positive such that $f\in\mathcal{F}^{M}$.

Claim: $(\phi(\frac{1}{P}.f(r(m))):\,m\in\omega)$ converges to $\phi(\frac{1}{P}.\chi_{\xi_{f}})$.

Proof.

Let $m\geq M$ be such that $P$ divides $Q_{m-1}$ and $\xi_{f}\in C\cap\operatorname{\text{supp }}\varrho^{r(m-1)}$. Then

$\phi(\frac{1}{P}.f(r(m)))=\phi(\frac{1}{P}.\sum_{\mu\in\operatorname{\text{supp }}f(r(m))}f(r(m))(\mu)\chi_{\mu})=$

$\phi(\sum_{\mu\in\operatorname{\text{supp }}f(r(m))}\frac{f(r(m))(\mu)}{P}.\chi_{\mu})=$ $\sum_{\mu\in\operatorname{\text{supp }}f(r(m))}\phi(\frac{f(r(m))(\mu)}{P}.\chi_{\mu})=$

$\sum_{\mu\in\operatorname{\text{supp }}f(r(m))}\frac{f(r(m))(\mu)}{P}.Q_{m}.\phi(\frac{1}{Q_{m}}.\chi_{\mu})=$

$\frac{1}{P}.Q_{m-1}\sum_{\mu\in\operatorname{\text{supp }}f(r(m))}K^{r(m-1)}_{r(m)}f(r(m))(\mu).\phi(\frac{1}{Q_{m}}.\chi_{\mu})\in$

$\frac{1}{P}.Q_{m-1}\sum_{\mu\in\operatorname{\text{supp }}f(r(m))}K^{r(m-1)}_{r(m)}f(r(m))(\mu).\varrho_{r(m)}(\mu)\subseteq$

$\frac{1}{P}.Q_{m-1}\varrho_{r(m-1)}(\xi_{f})$.

This last set is a neighborhood of $\phi(\frac{1}{P}.\chi_{\xi_{f}})$ and has length at most $\frac{1}{2^{r(m-2)}+1}$.

This proves the claim. ∎

Since $r[\omega]\in p$ it follows that the $p$-limit of $(\phi(\frac{1}{P}.f(n)):\,n\in\omega)$ is $\phi(\frac{1}{P}.\chi_{\xi_{f}})$.∎

5. Defining rational stacks

We define stacks as a tool to solve a system of arc equations. The way the rational stack is constructed is motivated by the construction in [8].

We want to solve arc equations related to the representatives of the basis for $G^{\omega}/p$. We construct a stack, associate the original arc equations to arc equations for the stack, solve the arc equations for the stack and these solutions will lead to a solution to the original system of arc equations.

For the most basic example, if we take a stack with a single element $\langle\{h\},\{Z_{n}:\,n\in A\}\rangle$, we choose a point $\zeta(n)$ (condition iii)) in the support of $h(n)$. The values of $h(n)(\zeta(n))$ and $Z_{n}$ and their ratio (condition v)) are the main ingredients to solve the arc equation. According to the size of the arc entry, either the denominator $Z_{n}$ or the numerator $h(n)(\zeta(n))$ must be large enough (condition v)). The output arc will shrink proportionally to the sizes of the numerator $Z_{n}$ and the denominator $h(n)(\zeta(n))$.

If we are dealing with a stack with more elements, the same $\zeta$ might have to be used for different sequences. Those $h(n)(\zeta(n))$ that increase at a proportional speed (condition vi)) will be solved together. But there may have other sequences using the same $\zeta$ that have different speeds.

The arc equations of the bricks whose denominator $h(n)(\zeta(n))$ is smaller will be solved first, since they require larger input arcs. To continue using the same $\zeta$ for arc equations for the next brick with the same $\zeta$, the $h(n)(\zeta(n))$ must be much larger to compensate that the output arcs have shrunk to solve the equations for the previous brick (condition vii)).

The sequences may use a different element of the support, but the shrinking of the arcs that solved previous equations must not interfere with the size of the arc related to a different point in the support. For this reason we need conditions ii), iv) and vii).

The idea to use the stacks to solve arc equations back and forth is based on the idea in [9].

The order used in the stack corresponds to the order we construct a stack associated to a finite number of elements of the representatives of a basis. The order to solve the arc equations is the reverse order of the stack.

Definition 5.1**.**

A rational stack is an nonuple $\langle\mathcal{B},\nu,\zeta,K,A,k_{0},k_{1},l,T\rangle$, where:

• •

$A\subseteq\omega$ is infinite,

• •

$k_{0}\leq k_{1}$ are natural numbers with $k_{1}>0$,

• •

$l:k_{1}\rightarrow\omega$,

• •

$\nu:k_{0}\rightarrow\kappa$,

• •

$\zeta:k_{1}\rightarrow\kappa^{\omega}$,

• •

$K:\omega\rightarrow\omega\setminus 2$ is such that for every $n\in A$, $n!T\mid K_{n}$,

• •

$\mathcal{B}=({{\mathcal{B}}}_{i,j}:\,i<k_{1},j<l_{i})$ is such that each $\mathcal{B}_{i,j}\subseteq H^{\omega}$ is finite,

• •

$T>0$ is an integer.

1. i)

$\zeta_{i}(n)=\nu_{i}$ for every $i<k_{0}$ and $n\in A$, 2. ii)

The elements $\nu_{i}\,(i<k_{0})$ and $\zeta_{j}(n)\,(k_{0}\leq j<k_{1},n\in A)$ are pairwise distinct, 3. iii)

$\zeta_{i}(n)\in\operatorname{\text{supp }}h(n)$, for each $i<k_{1}$, $j<l_{i}$, $h\in{\mathcal{B}}_{i,j}$ and $n\in A$, 4. iv)

$\zeta_{i}(n)\notin\operatorname{\text{supp }}h(n)$, for each, $i<i_{*}<k_{1}$, $j<l_{i_{*}}$ and $h\in{\mathcal{B}}_{i_{*},j}$ and $n\in A$, 5. v)

$\left(\frac{h(n)(\zeta_{i}(n))}{K_{n}}\right)_{n\in A}$ converges monotonically to $+\infty$, $-\infty$ or a real number, for each $i<k_{1}$, $j<l_{i}$ and $h\in{\mathcal{B}}_{i,j}$, 6. vi)

For every $i<k_{1},j<l_{i}$, there exists $h_{*}\in\mathcal{B}_{i,j}$ such that for every $h\in\mathcal{B}_{i,j}$, $\left(\frac{h(n)(\zeta_{i}(n))}{h_{*}(n)(\zeta_{i}(n))}\right)_{n\in A}$ converges to a real number $\theta_{h_{*}}^{h}$ and $(\theta_{h_{*}}^{h}:\,h\in{\mathcal{B}}_{i,j})$ is linearly independent (as a $\mathbb{Q}$-vector space), 7. vii)

For each $i<k_{1}$, $j^{\prime}<j<l_{i}$, $h\in{\mathcal{B}}_{i,j}$ and $h^{\prime}\in{\mathcal{B}}_{i,j^{\prime}}$, $\left(\frac{h(n)(\zeta_{i}(n))}{h^{\prime}(n)(\zeta_{i}(n))}\right)_{n\in A}$ converges monotonically to [math], 8. viii)

For each $i<k_{0}$ there exists $j<l_{i}$ such that $\frac{K}{T}.\chi_{\vec{\nu_{i}}}\in{\mathcal{B}}_{i,j}$, 9. ix)

$\left(|h(n)(\zeta_{i}(n))|\right)_{n\in A}$ is strictly increasing, for each $i<k_{1},j<l_{i}$ and $h\in{\mathcal{B}}_{i,j}$, and 10. x)

For each $i<k_{1},j<l_{i}$ and distinct $h,h_{*}\in{\mathcal{B}}_{i,j}$, either

• •

$|h(n)(\zeta_{i}(n))|>|h_{*}(n)(\zeta_{i}(n))|$ for each $n\in A$, or

• •

$|h(n)(\zeta_{i}(n))|=|h_{*}(n)(\zeta_{i}(n))|$ for each $n\in A$, or

• •

$|h(n)(\zeta_{i}(n))|<|h_{*}(n)(\zeta_{i}(n))|$ for each $n\in A$. 11. xi)

for all $\mu\in\kappa$, $i\in\omega$ such that $k_{0}\leq i<k_{1}$ and $g\in\bigcup_{j<l_{i}}B_{i,j}$, if $\{n\in\omega:\mu\in\operatorname{\text{supp }}g(n)(\mu)\}$ then $\left(\frac{g(n)(\mu)}{K_{n}}\right)_{n\in A}$ is constant.

The family ${\mathcal{B}}_{i,j}$ is named $(i,j)$-brick. This concept is inspired on the concepts defined in [8].

Notice that vi) implies that for every $i<k_{1}$ $j<l_{i}$, $h_{*}\in\mathcal{B}_{i,j}$, for every $h\in\mathcal{B}_{i,j}$, $\left(\frac{h(n)(\zeta_{i}(n))}{h_{*}(n)(\zeta_{i}(n))}\right)_{n\in A}$ converges to a real number $\theta_{h_{*}}^{h}$ and $(\theta_{h_{*}}^{h}:\,h\in{\mathcal{B}}_{i,j})$ is linearly independent.

6. Constructing a sequence of rational stacks

Given a finite sequence of functions we want to find an element of the ultrafilter $p$ that makes the restricted functions closer to the properties we want for the stack.

Lemma 6.1**.**

Suppose that ${\mathcal{G}}$ is a finite subset $G^{\omega}$, $p$ is a selective ultrafilter and $C\in p$. Suppose $\zeta,\zeta_{*}\in\kappa^{\omega}$ are such that there exist $g_{*}\in{\mathcal{G}}$ such that $\{n\in C:\,\zeta(n)\in\operatorname{\text{supp }}g_{*}(n)\}\in p$ and $\{n\in C:\,\zeta(n)=\zeta_{*}(n)\}\notin p$.

Then there exist $B^{\prime}\in p$, $B^{\prime}\subseteq C$ and $\mathcal{H}\subseteq\mathcal{G}$ such that:

1. ($\star$1)

$(\zeta(n))_{n\in B^{\prime}}$ is either constant or 1-1, 2. ($\star$2)

for each $g\in{\mathcal{G}}$, either $\zeta(n)\in\operatorname{\text{supp }}g(n)$ for each $n\in B^{\prime}$ or $\zeta(n)\notin\operatorname{\text{supp }}g(n)$ for each $n\in B^{\prime}$, 3. ($\star$3)

${\mathcal{H}}=\{g\in{\mathcal{G}}:\forall n\in B^{\prime}\,\zeta(n)\in\operatorname{\text{supp }}g(n)\}$ is nonempty, 4. ($\star$4)

$(g(n)(\zeta(n)))_{n\in B^{\prime}}$ converges strictly monotonically to an element of the extended real line, or it is constant and equal to a rational number, for each $g\in{\mathcal{H}}$, 5. ($\star$5)

given $f,g\in{\mathcal{H}}$ either $|g(n)(\zeta(n))>f(n)(\zeta(n))|$ for each $n\in B^{\prime}$, $|g(n)(\zeta(n))|=|f(n)(\zeta(n))|$ for each $n\in B^{\prime}$ or $|g(n)(\zeta(n))|<|f(n)(\zeta(n))|$ for each $n\in B^{\prime}$, 6. ($\star$6)

for each pair $g,h\in{\mathcal{H}}$, the sequence $(\frac{g(n)(\zeta(n))}{h(n)(\zeta(n))})_{n\in B^{\prime}}$ converges to $+\infty$, $-\infty$ or to a real number, and 7. ($\star$7)

$\zeta(n)\neq\zeta_{*}(m)$ for each $n,m\in B^{\prime}$.

Proof.

Everything follows from the selectivity of $p$. ∎

Notice that if $B\in p$ is such that $B\subseteq B^{\prime}$, then $(\star 1)-(\star 8)$ also hold for $B$.

Lemma 6.2**.**

Suppose that ${\mathcal{G}}$, $C$, $p$, $\zeta$, $\zeta_{*}$, $B^{\prime}$ and $\mathcal{H}$ be as in Lemma 6.1.

Suppose $g_{\#}\in\mathcal{H}$ is such that for every $g\in\mathcal{H}$, $(\frac{g(n)(\zeta(n))}{g_{\#}(n)(\zeta(n))})_{n\in B^{\prime}}$ converges to a real number (or, equivalently, is bounded).

Then there exist $B\in p$, with $B\subseteq B^{\prime}$, $\mathcal{B}\subseteq\mathcal{H}$, $\sigma:\mathcal{H}\setminus\mathcal{B}\rightarrow G^{\omega}$ and a family of real numbers $(\theta_{g_{\#},g}:g\in\mathcal{H})$ such that:

1. ($\star$9)

$g_{\#}\in\mathcal{B}$, 2. ($\star$10)

$(\frac{g(n)(\zeta(n))}{g_{\#}(n)(\zeta(n))})_{n\in B}$ converges to $\theta_{g_{\#},g}$, for every $g\in{\mathcal{H}}$, 3. ($\star$11)

$(\theta_{g_{\#},g}:\,g\in{\mathcal{B}})$ is a linearly independent set that generates the same $\mathbb{Q}$-vector space as $(\theta_{g_{\#},g}:g\in\mathcal{H})$, 4. ($\star$12)

for each $g\in\mathcal{H}\setminus\mathcal{B}$, $\mathcal{B}\cup\{g\}$ and $\mathcal{B}\cup\{\sigma(g)\}$ generate the same $\mathbb{Q}$-vector subspace of $G^{\omega}$, 5. ($\star$13)

for each $g\in\mathcal{H}\setminus\mathcal{B}$ and $h\in\mathcal{B}$, $\left(\frac{\sigma(g)(n)(\zeta(n))}{h(n)(\zeta(n))}\right)_{n\in B}$ converges to [math], 6. ($\star$14)

If $g\in\mathcal{H}\setminus\mathcal{B}$ and $\theta_{g_{\#},g}=0$, then $\sigma(g)=g$. 7. ($\star$15)

If $\zeta$ is constant and equal to $\nu$, $\mathcal{B}=\{\chi_{\vec{\nu}}\}$ and there exists $g^{*}\in\mathcal{H}$ such that $(g^{*}(n)(\nu))_{n\in B}$ is not constant mod $p$, then $\{n\in\omega:\nu\in\operatorname{\text{supp }}\sigma(g^{*}(n))\}\in p$.

Proof.

Consider $\{\theta_{g_{\#},g}:\,g\in{\mathcal{H}}\}$ as a subset of the ${\mathbb{Q}}$-vector space ${\mathbb{R}}$ and take ${\mathcal{B}}\subseteq{\mathcal{H}}$ containing $g_{\#}$ such that $(\theta_{g_{\#},h}:\,h\in{\mathcal{B}})$ is a basis for the vector space generated by $\{\theta_{g_{\#},g}:\,g\in{\mathcal{H}}\}$.

For the existence of $\sigma$, define $(r_{g,h}:g\in\mathcal{H},h\in\mathcal{B})$ by the expressions $\theta_{g_{\#},g}=\sum_{h\in\mathcal{B}}r_{g,h}\theta_{g_{\#},h}$. Now define $\sigma(g)=g-\sum_{h\in{\mathcal{B}}}r_{g,h}h$ for each $g\in\mathcal{H}\setminus\mathcal{B}$.

∎

Lemma 6.3**.**

Let $B\in p$. Suppose that $\zeta\in\kappa^{\omega}$, $m\in\omega$, $\zeta_{i}\in\kappa^{\omega}$ for $i<m$, are such that $\{n\in B:\forall i<m\;\zeta(n)\neq\nu_{i}\}\in p$. Suppose $\mathcal{G}$ is a finite subset of $G^{\omega}$ whose elements are distinct mod $p$, none of them is constant mod $p$ and such that $\{[f]_{p}:f\in\mathcal{G}\}\cup\{[\chi_{\vec{\xi}}]_{p}:\xi\in\kappa\}$ is linearly independent and there exists $g\in\mathcal{G}$ such that $\{n\in\omega:\zeta(n)\in\operatorname{\text{supp }}g(n)\}\in p$.

If $\zeta$ is constant, let $\nu$ be its value.

Then there exist finite $\mathcal{G}^{\prime}\subset G^{\omega}$, $l\in\omega\setminus\{0\}$, finite non-empty $\mathcal{B}_{j}\subset G^{\omega}$ for each $j<l$, $A\subset B$ such that:

1. (1)

For every $g\in\mathcal{G}^{\prime}$, $\{n\in\omega:\zeta(n)\in\operatorname{\text{supp }}g(n)\}\notin p$ and $\{n\in\omega:\nu_{i}\in\operatorname{\text{supp }}g(n)\}\notin p$ for each $i<m$, 2. (2)

$\mathcal{B}_{j}\cap\mathcal{B}_{j^{\prime}}=\varnothing$ for $j\neq j^{\prime}$, $\mathcal{B}_{j}\cap\mathcal{G}^{\prime}=\varnothing$ for each $j<l$ and $\{[f]_{p}:f\in\mathcal{G}^{\prime}\cup\bigcup_{j<l}\mathcal{B}_{j}\}$ is a linearly independent subset of $G^{\omega}/p$. Also, if $f,h\in\mathcal{G}^{\prime}\dot{\cup}\dot{\bigcup}_{k<l}\mathcal{B}_{k}$ are distinct, then $[f]_{p}\neq[h]_{p}$. 3. (3)

As vector subspaces of $G^{\omega}$, $\langle\mathcal{G}^{\prime}\cup\bigcup_{j<l}\mathcal{B}_{j}\rangle=\langle\mathcal{G}\cup\{\chi_{\vec{\nu}}\}\rangle$ if $\zeta$ is constant and $\langle\mathcal{G}^{\prime}\cup\bigcup_{j<l}\mathcal{B}_{j}\rangle=\langle\mathcal{G}\rangle$ otherwise, 4. (4)

$\zeta(n)\in\operatorname{\text{supp }}h(n)$, for each $j<l$, $h\in{\mathcal{B}}_{j}$ and $n\in A$, 5. (5)

$\nu_{i}\notin\operatorname{\text{supp }}h(n)$, for each $i<m$, $j<l$ and $h\in{\mathcal{B}}_{j}$ and $n\in A$, 6. (6)

$(h(n)(\zeta(n)))_{n\in A}$ converges strictly monotonically to an element of the extended real line, or it is constant and equal to a rational number, for each $j<l$ and $h\in{\mathcal{B}}_{j}$, 7. (7)

For every $j<l$, there exists $h_{*}\in\mathcal{B}_{j}$ such that for every $h\in\mathcal{B}_{j}$, $\left(\frac{h(n)(\zeta(n))}{h_{*}(n)(\zeta(n))}\right)_{n\in A}$ converges to a real number $\theta_{h_{*}}^{h}$ and $(\theta_{h_{*}}^{h}:\,h\in{\mathcal{B}}_{j})$ is linearly independent (as a $\mathbb{Q}$-vector space), 8. (8)

For each $j^{\prime}<j<l$, $h\in{\mathcal{B}}_{j}$ and $h^{\prime}\in{\mathcal{B}}_{j^{\prime}}$ $\left(\frac{h(n)(\zeta(n))}{h^{\prime}(n)(\zeta(n))}\right)_{n\in A}$ converges monotonically to [math], 9. (9)

If $\zeta$ is constant, there exists $j<l$ such that $\chi_{\vec{\nu}}\in{\mathcal{B}}_{j}$, 10. (10)

For each $j<l$ and distinct $h,h^{\prime}\in{\mathcal{B}}_{j}$, either

• •

$|h(n)(\zeta(n))|>|h^{\prime}(n)(\zeta(n))|$ for each $n\in A$, or

• •

$|h(n)(\zeta(n))|=|h^{\prime}(n)(\zeta(n))|$ for each $n\in A$, or

• •

$|h(n)(\zeta(n))|<|h^{\prime}(n)(\zeta(n))|$ for each $n\in A$. 11. (11)

No element of $\mathcal{G}^{\prime}$ is constant mod $p$ and $\{[f]_{p}:f\in\mathcal{G}^{\prime}\}\cup\{[\chi_{\vec{\xi}}]_{p}:\xi\in\kappa\}$ is linearly independent, 12. (12)

if $i<m$ and $n,n^{\prime}\in A$ are distinct, then $\zeta(n)\neq\zeta_{i}(n^{\prime})$, and 13. (13)

$|\mathcal{G}^{\prime}|<|\mathcal{G}|$.

Proof.

Since $p$ is selective, we may suppose by shrinking $B$ is necessary that if $i<m$ and $n,n^{\prime}\in B$ are distinct, then $\zeta(n)\neq\zeta_{i}(n^{\prime})$. Clearly, this property will hold for any subset of $B$.

If $\zeta$ is constant, let $\mathcal{G}_{0}=\mathcal{G}\cup\{\chi_{\vec{\nu}}\}$. If not, let $\mathcal{G}_{0}=\mathcal{G}$. We will construct:

• •

$A_{j}\in p$ for $j\in\omega$, with $A_{j}\subseteq B$,

• •

$\mathcal{G}_{j}\subseteq G^{\omega}$ for $j\in\omega$,

• •

$\mathcal{H}_{j}\subseteq{\mathcal{G}}_{j}$ for $j\in\omega$,

• •

${\mathcal{B}}_{j}\subseteq\mathcal{H}_{j}$ for $j\in\omega$,

• •

$\sigma_{j}:\mathcal{H}_{j}\setminus{\mathcal{B}}_{j}\rightarrow G^{\omega}$ for $j\in\omega$,

satisfying:

1. i)

$A_{j}\subseteq A_{j-1}$ for every $j<l$, 2. ii)

for each $j<l$, $\{[f]_{p}:f\in\mathcal{G}_{j}\dot{\cup}\dot{\bigcup}_{k<j}\mathcal{B}_{k}\}$ is a linearly independent subset of $G^{\omega}/p$. Also, if $f,h\in\mathcal{G}_{j}\dot{\cup}\dot{\bigcup}_{k<j}\mathcal{B}_{k}$ are distinct, then $[f]_{p}\neq[h]_{p}$. 3. iii)

if $j\in\omega$, $h\in{\mathcal{H}}_{j}$ and $n\in A_{j}$, then $\zeta(n)\in\operatorname{\text{supp }}h(n)$, 4. iv)

if $j\in\omega$, $h\in{\mathcal{G}}_{j}\setminus{\mathcal{H}}_{j}$ and $n\in A_{j}$, then $\zeta(n)\notin\operatorname{\text{supp }}h(n)$, 5. v)

if $j\in\omega$, $h\in{\mathcal{H}}_{j}$, $i<m$ and $n\in A_{j}$, then $\nu_{i}\notin\operatorname{\text{supp }}h(n)$, 6. vi)

if $j\in\omega$ and $h\in{\mathcal{H}}_{j}$, then $(h(n)(\zeta(n)))_{n\in A_{j}}$ converges strictly monotonically to an element of the extended real line, or it is constant and equal to a rational number, 7. vii)

for every $j\in\omega$, $\mathcal{B}_{j}\neq\varnothing$ iff $\exists g\in\mathcal{G}_{j}\,\{n\in\omega:\zeta(n)\in\operatorname{\text{supp }}g(n)\}\in p$, 8. viii)

for every $j<\omega$, if $\mathcal{B}_{j}\neq\varnothing$, then there exists $h_{*}\in\mathcal{B}_{j}$ such that for every $h\in\mathcal{B}_{j}$, $\left(\frac{h(n)(\zeta(n))}{h_{*}(n)(\zeta(n))}\right)_{n\in A_{j}}$ converges to a real number $\theta_{h_{*}}^{h}$ and $(\theta_{h_{*}}^{h}:\,h\in{\mathcal{B}}_{j})$ is linearly independent (as a $\mathbb{Q}$-vector space), 9. ix)

for every $j,j^{\prime}\in\omega$, if $j^{\prime}<j$, $h\in{\mathcal{B}}_{j}$ and $h^{\prime}\in{\mathcal{B}}_{j^{\prime}}$, then $\left(\frac{h(n)(\zeta(n))}{h^{\prime}(n)(\zeta(n))}\right)_{n\in A_{j}}$ converges monotonically to [math], 10. x)

if $\zeta$ is constant, there exists $j\in\omega$ such that $\chi_{\vec{\nu}}\in{\mathcal{B}}_{j}$, 11. xi)

if $j\in\omega$ and $h,h^{\prime}\in{\mathcal{B}}_{j}$, either

• •

$|h(n)(\zeta(n))|>|h^{\prime}(n)(\zeta(n))|$ for each $n\in A_{j}$, or

• •

$|h(n)(\zeta(n))|=|h^{\prime}(n)(\zeta(n))|$ for each $n\in A_{j}$, or

• •

$|h(n)(\zeta(n))|<|h^{\prime}(n)(\zeta(n))|$ for each $n\in A_{j}$. 12. xii)

for each $j\in\omega$ and $g\in\mathcal{H}_{j}$, $(|g(n)(\zeta(n))|)_{n\in A_{j}}$ is either constant or strictly increasing, 13. xiii)

for each $j\in\omega$ and $g\in\mathcal{H}_{j}\setminus\mathcal{B}_{j}$, $\mathcal{B}_{j}\cup\{g\}$ and $\mathcal{B}_{j}\cup\{\sigma(g)\}$ generate the same $\mathbb{Q}$-vector subspace of $G^{\omega}$, 14. xiv)

for each $j\in\omega$, $g\in\mathcal{G}_{j}$ and $i<m$, $\{n\in\omega:\nu_{i}\in\operatorname{\text{supp }}g(n)\}\notin p$, 15. xv)

for each $j,j^{\prime}\in\omega$, $\mathcal{G}_{j}\cup\bigcup_{k<j}\mathcal{B}_{k}$ generates the same subspace of $G^{\omega}$ as $\mathcal{G}_{j^{\prime}}\cup\bigcup_{k<j^{\prime}}\mathcal{B}_{k}$ 16. xvi)

if $\zeta$ is constant, then for each $j\in\omega$, $\chi_{\vec{\nu}}\in\bigcup_{k<j}\mathcal{B}_{j}\cup\mathcal{G}_{j}$, 17. xvii)

$\mathcal{G}_{j+1}=(\mathcal{G}_{j}\setminus\mathcal{H}_{j})\cup\operatorname{ran}\sigma_{j}$, and 18. xviii)

if $\zeta$ is constant, $j\in\omega$ and $\chi_{\vec{\nu}}\in\mathcal{H}_{j}\setminus\mathcal{B}_{j}$, then $\sigma_{j}(\chi_{\vec{\nu}})=\chi_{\vec{\nu}}$. 19. xix)

if $\zeta$ is constant and $\mathcal{B}_{0}=\{\chi_{\vec{\nu}}\}$, then there exists $g\in\mathcal{H}_{0}\setminus\mathcal{B}_{0}$ such that $\{n\in\omega:\nu\in\operatorname{\text{supp }}\sigma_{0}(g)\}\in p$.

Suppose we have carried on such a recursion. By ii) and xviii), one of the $\mathcal{B}_{j}$’s must be empty. Let $l$ be the first $j$ such that $\mathcal{B}_{j}=\varnothing$. By vii), $\forall g\in\mathcal{G}_{l},\{n\in\omega:\zeta(n)\notin g(n)\}\in p$. Since there exists $g\in\mathcal{G}$ such that $\{n\in\omega:\zeta(n)\in\operatorname{\text{supp }}g(n)\}\in p$, it follows that $j>0$. Let $A=A_{l-1}$, $\mathcal{G}^{\prime}=\mathcal{G}_{l}$. Notice that every $\mathcal{B}_{j}$ is nonempty for $j<l$.

(1) holds by the previous observation, i), v) and by the fact that $\mathcal{B}_{j}\subseteq\mathcal{H}_{j}$. (2) holds by (ii). (3) follows from xvii) using $j=l$, $j^{\prime}=0$. (4)-(8), (10) and (11) follow easily from (i), (iii)-(vii), (viii), (ix), (xi) and (xii). Suppose $(9)$ doesn’t hold. Then by xviii), $\chi_{\vec{\nu}}\in\mathcal{G}_{l}$. But then, by vii), $\mathcal{B}_{l}\neq\varnothing$, a contradiction.

(13) holds by ii), because $\langle\mathcal{G}^{\prime}\rangle\subseteq\langle\mathcal{G}_{0}\rangle$ and because, if $\zeta$ is constant then, by xviii) and (9), $\mathcal{G}^{\prime}\cup\{\chi_{\vec{\nu}}\}$ is linearly independent.

(14) holds: if $\zeta$ is not constant, it follows from (2) and (3). If it is constant, first, notice that, by xxi), xix) for $j=0$, and vii) for $j=1$, it follows that $l>1$ or $\mathcal{B}_{0}\neq\{\chi_{\vec{\nu}}\}$. Either way, $\mathcal{B}=\bigcup_{i<l}\mathcal{B}_{i}\setminus\{\chi_{\vec{\nu}}\}$ is nonempty. By (2), (3) and (9), analyzing dimensions it follows that $1+|\mathcal{B}|+|\mathcal{G}^{\prime}|=1+|\mathcal{G}|$, so $|\mathcal{G}^{\prime}|<|\mathcal{G}|$.

Construction: For step [math], $\mathcal{G}_{0}$ is already defined. We apply Lemma 6.1 $m$ times using $\zeta(n)=\zeta(n)$ and $\zeta_{*}(n)=\nu_{i}$ for every $n$. If $m=0$ we apply it once using $\zeta_{*}(n)=\zeta(n)^{\prime}$ for every $n$ for some $\zeta(n)^{\prime}\neq\zeta(n)$. We now have $\mathcal{H}_{0}$ and $A^{\prime}_{0}\subset B$.

If it is the case that $(h(n)(\zeta(n)))_{n\in A_{0}}$ converges to a real number for every $h\in\mathcal{H}_{0}$ and that $\zeta$ is constant, then we apply Lemma 6.2 with $g_{\#}=\chi_{\vec{\nu}}$, and obtain $A_{0}\subset A^{\prime}_{0}$, $\mathcal{B}_{0}\subset\mathcal{H}_{0}$ and $\sigma_{0}$. If not, then we take any $g_{\#}\in\mathcal{H}_{0}$ that satisfies the hypothesis of Lemma 6.2 – one does exist because of ($\star 5$), which also implies that for such a $g_{\#}$, $\left(\frac{1}{g_{\#}(n)(\zeta(n))}\right)_{n\in A_{0}}$ converges to 0, and thus in case $\zeta$ is constant, $\sigma_{0}(\chi_{\vec{\nu}})=\chi_{\vec{\nu}}$. Either way, we obtain $\sigma_{0}$, $\mathcal{B}_{0}$ and $\mathcal{A}_{0}$. It is straightforward to verify that $(i)-(xxi)$ hold for this step.

For the inductive step, we define $\mathcal{G}_{j+1}$ as in xix). If $\nexists g\in\mathcal{G}_{j+1}\,\{n\in\omega:\zeta(n)\in\operatorname{\text{supp }}g(n)\}\in p$, then we define $\mathcal{H}_{j+1}=\varnothing$, $A_{j+1}\subseteq A_{j}$ satisfying v) with $j$ swapped by $j+1$ and $\mathcal{B}_{j+1}=\sigma_{j+1}=\varnothing$. If not, we proceed as in step [math]: we first apply Lemma 6.1 to obtain $\mathcal{H}_{j+1}$ and $A^{\prime}_{j+1}\subset A_{j}$ and then similarly apply Lemma 6.2 to obtain $\mathcal{B}_{j+1}$, $A_{j+1}$ and $\sigma_{j+1}$. It is straightforward to verify that (i)-(xix) hold for this step.

∎

Lemma 6.4**.**

Suppose $\mathcal{G}$ is a finite subset of $G^{\omega}$ whose elements are distinct mod $p$ and none of them is constant mod $p$ such that $\{[f]_{p}:f\in\mathcal{G}\}\cup\{[\chi_{\vec{\nu}}]_{p}:\nu\in\kappa\}$ is a linearly independent subset of $G^{\omega}/p$. Then: either there exist $\mu\in\kappa$, $g^{*}\in\mathcal{G}$ and $A\in p$ such that $(g^{*}(n)(\mu))_{n\in A}$ is one-to-one, or there exists $A\in p$ such that for every $g\in\mathcal{G}$ there exists $\zeta_{g}\in\kappa^{\omega}$ satisfying $\zeta_{g}(n)\in\operatorname{\text{supp }}g(n)$ for all $n\in A$ and $\zeta_{g}|A$ is one-to-one.

Proof.

Suppose that for all $\mu\in\kappa$, for all $g\in\mathcal{G}$ and for all $A\in p$, $(g(n)(\mu))_{n\in A}$ is not one-to-one. Then by the selectivity of $p$, for all $\mu\in\kappa$ there exist $B_{\mu}\in p$ such that for all $g\in\mathcal{G}$, $(g(n)(\mu))_{n\in B_{\mu}}$ is constant.

Fix a $g\in\mathcal{G}$. By selectivity, there exists $B\in p$ such that either the sequence $(|\operatorname{\text{supp }}g(n)|)_{n\in B}$ is strictly increasing or it is constant. If it is strictly increasing, then we may pick recursively $\tilde{\zeta}_{g}(n)\in\operatorname{\text{supp }}g(n)$ for each $n\in B$ in a way that $\tilde{\zeta}_{g}$ is one-to-one. Define $A_{g}=B$.

If it is constant, let $k\in\omega$ be that constant. Since $g$ is not 0 mod $p$, $k\geq 1$. For each $i<k$, let $w_{i}\in\kappa^{\omega}$ be such that $\operatorname{\text{supp }}g(n)=\{w_{0}(n),\ldots,w_{k-1}(n)\}$ for each $n\in B$. Then there exists $C\subset B$, $C\in p$ such that for each $i<k$, $(w_{i}(n))_{n\in C}$ is one-to-one or constant.

We claim that there is a $j<k$ such that $(w_{j}(n))_{n\in C}$ is one-to-one. Suppose all of them are constant; take $\mu_{j}$ for each $j<k$ so that $w_{j}(n)=\mu_{j}$ for all $n\in C$. Then, since $(g(n)(\mu_{j}))_{n\in B_{\mu_{j}}}$ is constant for each $j<k$, let $r_{j}$ be those constants. Let $D=C\cap\bigcap_{j<k}B_{\mu_{j}}$. We have that $D\in p$ and $g(n)=\left(\sum_{j<k}r_{j}\chi_{\overrightarrow{\mu_{j}}}\right)(n)$ for all $n\in D$, and so $[g]_{p}=\sum_{j<k}r_{j}[\chi_{\overrightarrow{\mu_{j}}}]_{p}$. This contradicts the hypothesis that $\{[f]_{p}:f\in\mathcal{G}\}\cup\{[\chi_{\vec{\nu}}]_{p}:\nu\in\kappa\}$ is a linearly independent subset of $G^{\omega}/p$.

Therefore, there exists a $j<k$ such that $(w_{j}(n))_{n\in C}$ is one-to-one, and so we define $\tilde{\zeta}_{g}=w_{j}$ and $A_{g}=C$.

Thus if we define $A=\bigcap_{g\in\mathcal{G}}A_{g}$ and $\zeta_{g}\in\kappa^{\omega}$ such that $\zeta_{g}|A=\tilde{\zeta}_{g}|A$, we have the desired result. ∎

Now we restate Lemma 3.1, which we are going to prove.

Lemma**.**

Let $B\in p$ and $\mathcal{G}$ be a finite subset of $G^{\omega}$ whose elements are distinct mod $p$ and none of them is constant mod $p$ such that $\{[f]_{p}:f\in\mathcal{G}\}\cup\{[\chi_{\vec{\nu}}]_{p}:\nu\in\kappa\}$ is linearly independent. Then there exist a rational stack $\mathcal{S}=\langle\mathcal{B},\nu,\zeta,K,A,k_{0},k_{1},l\rangle$ and a positive integer $S$ such that, by defining ${\mathcal{A}}={\mathcal{G}}\cup\{\chi_{\vec{\nu_{i}}}:\,i<k_{0}\}$ and ${\mathcal{C}}=\frac{\bigcup_{i<k_{1},j<l_{i}}{\mathcal{B}}_{i,j}}{{K}}$, there exist ${\mathcal{M}}:\mathcal{A}\times\mathcal{C}\rightarrow\mathbb{Z}$, ${\mathcal{N}}:\mathcal{C}\times\mathcal{A}\rightarrow\mathbb{Z}$ satisfying:

1. (1)

$\{[f]_{p}:\,f\in{\mathcal{A}}\}$ and $\{[h]_{p}:\,h\in{\mathcal{C}}\}$ generate the same subspace; 2. (2)

$f(n)=\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}h(n)$, for each $n\in A$ and $f\in{\mathcal{A}}$, 3. (3)

$h(n)=\frac{1}{S}.\sum_{f\in{\mathcal{A}}}{\mathcal{N}}_{h,f}f(n)$, for each $n\in A$ and $h\in{\mathcal{C}}$, 4. (4)

$K\mathcal{A}\subseteq H^{\omega}$, 5. (5)

$K\mathcal{C}\subseteq H^{\omega}$, and 6. (6)

$A\subseteq B$.

Proof.

(of Lemma 3.1) We will start building a sequence that will almost be the stack $\mathcal{S}$ which we will associate with $\mathcal{G}$.

Claim: There exist:

• •

$A^{\prime}\in p$ with $A^{\prime}\subseteq B$,

• •

$k_{0}\in\omega$

• •

$l^{\prime}:k_{0}\rightarrow\omega\setminus\{0\}$,

• •

$\nu:k_{0}\rightarrow\kappa$,

• •

$\zeta^{\prime}:k_{0}\rightarrow\kappa^{\omega}$,

• •

$\mathcal{G}^{\prime}\subset G^{\omega}$,

• •

$(\hat{\mathcal{B}}_{i,j}:i<k_{0},j<l_{i})$ a family of nonempty subsets of $G^{\omega}$,

Satisfying:

1. (1)

$\zeta^{\prime}_{i}(n)=\nu_{i}$ for every $i<k_{0}$ and $n\in A^{\prime}$, 2. (2)

The elements $\nu_{i}\,(i<k_{0})$ are pairwise distinct, 3. (3)

$\nu_{i}\in\operatorname{\text{supp }}h(n)$, for each $i<k_{0}$, $j<l_{i}$, $h\in\hat{\mathcal{B}}_{i,j}$ and $n\in A$, 4. (4)

$\nu_{i}\notin\operatorname{\text{supp }}h(n)$, for each, $i<i_{*}<k_{0}$, $j<l_{i_{*}}$ and $h\in\hat{\mathcal{B}}_{i_{*},j}$ and $n\in A$, 5. (5)

$(h(n)(\nu_{i}))_{n\in A}$ converges strictly monotonically to an element of the extended real line, or it is constant and equal to a rational number, for each $i<k_{0}$, $j<l_{i}$ and $h\in\hat{\mathcal{B}}_{i,j}$, 6. (6)

For every $i<k_{0},j<l_{i}$, there exists $h_{*}\in\hat{\mathcal{B}}_{i,j}$ such that for every $h\in\hat{\mathcal{B}}_{i,j}$, $\left(\frac{h(n)(\nu_{i})}{h_{*}(n)(\nu_{i})}\right)_{n\in A}$ converges to a real number $\theta_{h_{*}}^{h}$ and $(\theta_{h_{*}}^{h}:\,h\in\hat{\mathcal{B}}_{i,j})$ is linearly independent (as a $\mathbb{Q}$-vector space), 7. (7)

For each $i<k_{0}$, $j^{\prime}<j<l_{i}$, $h\in\hat{\mathcal{B}}_{i,j}$ and $h^{\prime}\in\hat{\mathcal{B}}_{i,j^{\prime}}$, $\left(\frac{h(n)(\nu_{i})}{h^{\prime}(n)(\nu_{i})}\right)_{n\in A}$ converges monotonically to [math], 8. (8)

$\left(|h(n)(\nu_{i})|\right)_{n\in A^{\prime}}$ is constant or strictly increasing, for each $i<k_{0},j<l_{i}$ and $h\in\hat{\mathcal{B}}_{i,j}$, 9. (9)

For each $i<k_{0}$ there exists $j<l_{i}$ such that $\chi_{\vec{\nu_{i}}}\in\hat{\mathcal{B}}_{i,j}$, 10. (10)

For each $i<k_{0},j<l_{i}$ and distinct $h,h_{*}\in\hat{\mathcal{B}}_{i,j}$, either

• •

$|h(n)(\nu_{i})|>|h_{*}(n)(\nu_{i})|$ for each $n\in A$, or

• •

$|h(n)(\nu_{i})|=|h_{*}(n)(\nu_{i})|$ for each $n\in A$, or

• •

$|h(n)(\nu_{i})|<|h_{*}(n)(\nu_{i})|$ for each $n\in A$. 11. (11)

for all $\mu\in\kappa$, for every $g\in\mathcal{G}^{\prime}$, if $\{n\in\omega:\mu\in\operatorname{\text{supp }}g(n)(\mu)\}\in p$ then $(g(n)(\mu))_{n\in A}$ is constant, 12. (12)

for all $g\in\mathcal{G}^{\prime}$ and all $i<k_{0}$, $\{n\in\omega:\nu_{i}\in\operatorname{\text{supp }}g(n)\}\notin p$, 13. (13)

If $i,i^{\prime}<k_{0}$, $j<l^{\prime}_{i}$, $j^{\prime}<l^{\prime}_{i^{\prime}}$ and $(i,j)\neq(i^{\prime},j^{\prime})$, then $\hat{\mathcal{B}}_{i,j}\cap\hat{\mathcal{B}}_{i^{\prime},j^{\prime}}=\varnothing$, $\hat{\mathcal{B}}_{i,j}\cap\mathcal{G}^{\prime}=\varnothing$ for each $j<l$ and $\{[f]_{p}:f\in\mathcal{G}^{\prime}\cup\bigcup_{i<k_{0}}\bigcup_{j<l_{i}}\hat{\mathcal{B}}_{i,j}\}$ is a linearly independent subset of $G^{\omega}/p$. Also, if $f,h\in\mathcal{G}^{\prime}\dot{\cup}\dot{\bigcup}_{i<k_{0}}\dot{\bigcup}_{j<l_{i}}\hat{\mathcal{B}}_{i,j}$ are distinct, then $[f]_{p}\neq[h]_{p}$. 14. (14)

As vector subspaces of $G^{\omega}$, $\langle\mathcal{G}^{\prime}\cup\bigcup_{i<k_{0}}\bigcup_{j<l_{i}}\hat{\mathcal{B}}_{i,j}\rangle=\langle\mathcal{G}\cup\{\chi_{\vec{\nu_{i}}}:i<k_{0}\}\rangle$ 15. (15)

No element of $\mathcal{G}^{\prime}$ is constant mod $p$ and $\{[f]_{p}:f\in\mathcal{G}^{\prime}\}\cup\{[\chi_{\vec{\xi}}]_{p}:\xi\in\kappa\}$ is linearly independent.

If $(11)$ already holds for $\mathcal{G}$ and $A=B$, we let $\mathcal{G}^{\prime}=\mathcal{G}$, $A^{\prime}=B$, $k_{0}=0$ and the other sequences be $\varnothing$.

If not, then we may take a $\nu_{0}\in\kappa$ such that there exist $g\in\mathcal{G}$ and $B^{\prime}\subset B$, $B^{\prime}\in p$ such that $\{n\in\omega:\nu_{0}\in\operatorname{\text{supp }}g(n)\}\in p$ and $(g(n)(\nu_{0}))_{n\in B^{\prime}}$ is one-to-one.

We define $\mathcal{G}_{0}=\mathcal{G}$ and apply Lemma 6.3 to $B^{\prime}$, $m=0$, $\mathcal{G}_{0}$, $\zeta(n)=\nu_{0}$ for all $n\in B^{\prime}$ and obtain $\mathcal{G}^{\prime}_{0}$, $A_{0}$, $l_{0}$ and $\hat{\mathcal{B}}_{0,j}$ for $j<l_{0}$ satisfying everything but (11) (possibly) by using $k_{0}=0$ .

Suppose the recursion has been done up to $m\in\omega$ and we have, for $i<m$, $\nu_{i}$ $\mathcal{G}^{\prime}_{i}$, $A_{i}$, $l_{i}$ and $\hat{\mathcal{B}}_{i,j}$ for $j<l_{i}$ satisfying everything but (10) (possibly) by using $k_{0}=m$. If (10) holds for $\mathcal{G}^{\prime}_{m-1}$, then we let $A^{\prime}=A_{m-1}$, $k_{0}=m$ and $\mathcal{G}^{\prime}=\mathcal{G}^{\prime}_{m-1}$ and the recursion is over. If not, we take $\nu_{m}\in\kappa$ such that there exist $g\in\mathcal{G}^{\prime}_{m-1}$ and $B^{\prime}\subset A_{m-1}$, $B^{\prime}\in p$ such that $\{n\in\omega:\nu_{m}\in\operatorname{\text{supp }}g(n)\}\in p$ and $(g(n)(\nu_{m}))_{n\in B^{\prime}}$ is one-to-one. Notice that item (11) implies that $\nu_{m}\neq\nu_{i}$ for every $i<m$. We then apply Lemma 6.3 to $B^{\prime}$, $m$, $\mathcal{G}^{\prime}_{m-1}$, $\zeta(n)=\nu_{m}$ for all $n\in B^{\prime}$, $\zeta_{i}(n)=\nu_{i}$ for all $n\in B^{\prime}$ and $i<m$, and obtain $\mathcal{G}^{\prime}_{m}$, $A_{m}$, $l_{m}$ and $\hat{\mathcal{B}}_{m,j}$ for $j<l_{m}$ satisfying everything but (11) (possibly) by using $k_{0}=m$.

The recursion must eventually stop due to items (12), (13) and the fact that the $\hat{\mathcal{B}}_{i,j}$’s are nonempty. We now have $A^{\prime}$, $k_{0}$, $l^{\prime}$, $\nu$, $\zeta^{\prime}$, $\mathcal{G}^{\prime}$ and $(\hat{\mathcal{B}}_{i,j}:i<k_{0},j<l_{i})$ as in the Claim above.

Claim: There exist:

• •

$A^{\prime\prime}\in p$ with $A^{\prime\prime}\subseteq A^{\prime}$,

• •

$k_{1}\in\omega\backslash\{0\}$

• •

$l:k_{1}\rightarrow\omega\setminus\{0\}$ extending $l^{\prime}$,

• •

$\zeta:k_{1}\rightarrow\kappa^{\omega}$ extending $\zeta^{\prime}$,

• •

$(\hat{\mathcal{B}}_{i,j}:i<k_{1},j<l_{i})$ a family of nonempty subsets of $G^{\omega}$,

Satisfying:

1. (1)

$\zeta_{i}(n)=\nu_{i}$ for every $i<k_{1}$ and $n\in A^{\prime\prime}$, 2. (2)

The elements $\nu_{i}\,(i<k_{0})$ and $\zeta_{j}(n)$ $(k_{0}\leq j<k_{1},n\in A)$ are pairwise distinct, 3. (3)

$\zeta_{i}(n)\in\operatorname{\text{supp }}h(n)$, for each $i<k_{1}$, $j<l_{i}$, $h\in\hat{\mathcal{B}}_{i,j}$ and $n\in A^{\prime\prime}$, 4. (4)

$\zeta_{i}(n)\notin\operatorname{\text{supp }}h(n)$, for each, $i<i_{*}<k_{1}$, $j<l_{i_{*}}$ and $h\in\hat{\mathcal{B}}_{i_{*},j}$ and $n\in A^{\prime\prime}$, 5. (5)

$(h(n)(\zeta_{i}(n)))_{n\in A^{\prime\prime}}$ converges strictly monotonically to an element of the extended real line, or it is constant and equal to a rational number, for each $i<k_{1}$, $j<l_{i}$ and $h\in\hat{\mathcal{B}}_{i,j}$, 6. (6)

For every $i<k_{1},j<l_{i}$, there exists $h_{*}\in\hat{\mathcal{B}}_{i,j}$ such that for every $h\in\hat{\mathcal{B}}_{i,j}$, $\left(\frac{h(n)(\zeta_{i}(n))}{h_{*}(n)(\zeta_{i}(n))}\right)_{n\in A^{\prime\prime}}$ converges to a real number $\theta_{h_{*}}^{h}$ and $(\theta_{h_{*}}^{h}:\,h\in\hat{\mathcal{B}}_{i,j})$ is linearly independent (as a $\mathbb{Q}$-vector space), 7. (7)

For each $i<k_{1}$, $j^{\prime}<j<l_{i}$, $h\in\hat{\mathcal{B}}_{i,j}$ and $h^{\prime}\in\hat{\mathcal{B}}_{i,j^{\prime}}$, $\left(\frac{h(n)(\zeta_{i}(n))}{h^{\prime}(n)(\zeta_{i}(n))}\right)_{n\in A^{\prime\prime}}$ converges monotonically to [math], 8. (8)

$\left(|h(n)(\zeta_{i}(n))|\right)_{n\in A^{\prime\prime}}$ is strictly increasing, for each $i<k_{1},j<l_{i}$ and $h\in\hat{\mathcal{B}}_{i,j}$, 9. (9)

For each $i<k_{0}$ there exists $j<l_{i}$ such that $\chi_{\vec{\nu_{i}}}\in\hat{\mathcal{B}}_{i,j}$, 10. (10)

For each $i<k_{1},j<l_{i}$ and distinct $h,h_{*}\in\hat{\mathcal{B}}_{i,j}$, either

• •

$|h(n)(\zeta_{i}(n))|>|h_{*}(n)(\zeta_{i}(n))|$ for each $n\in A^{\prime\prime}$, or

• •

$|h(n)(\zeta_{i}(n))|=|h_{*}(n)(\zeta_{i}(n))|$ for each $n\in A^{\prime\prime}$, or

• •

$|h(n)(\zeta_{i}(n))|<|h_{*}(n)(\zeta_{i}(n))|$ for each $n\in A^{\prime\prime}$. 11. (11)

for all $\mu\in\kappa$, for every $i\geq k_{0}$, $j<l_{i}$ and $g\in\hat{\mathcal{B}}_{i,j}$, if $\{n\in\omega:\mu\in\operatorname{\text{supp }}g(n)(\mu)\}\in p$ then $(g(n)(\mu))_{n\in A^{\prime\prime}}$ is constant, 12. (12)

As vector subspaces of $G^{\omega}$, $\langle\bigcup_{i<k_{0}}\bigcup_{j<l_{i}}\hat{\mathcal{B}}_{i,j}\rangle=\langle\mathcal{G}\cup\{\chi_{\vec{\nu_{i}}}:i<k_{0}\}\rangle$, 13. (13)

If $i,i^{\prime}<k_{1}$, $j<l^{\prime}_{i}$, $j^{\prime}<l^{\prime}_{i^{\prime}}$ and $(i,j)\neq(i^{\prime},j^{\prime})$, then $\hat{\mathcal{B}}_{i,j}\cap\hat{\mathcal{B}}_{i^{\prime},j^{\prime}}=\varnothing$, and $\{[f]_{p}:f\in\bigcup_{i<k_{0}}\bigcup_{j<l_{i}}\hat{\mathcal{B}}_{i,j}\}$ is a linearly independent subset of $G^{\omega}/p$. Also, if $f,h\in\bigcup_{i<k_{0}}\bigcup_{j<l_{i}}\hat{\mathcal{B}}_{i,j}$ are distinct, then $[f]_{p}\neq[h]_{p}$.

For the initial step $k_{0}$ of the recursion, we first notice that by item (11) of the previous Claim, applying Lemma 6.4 we may take $A^{\prime\prime}_{k_{0}}\in p$ such that for every $g\in\mathcal{G}^{\prime}$ there exists $\zeta_{g}\in\kappa^{\omega}$ such that $\zeta_{g}(n)\in\operatorname{\text{supp }}g(n)$ for all $n\in A^{\prime\prime}_{k_{0}}$ and $\zeta_{g}|A^{\prime\prime}_{k_{0}}$ is one-to-one. Define $\mathcal{G}_{k_{0}}=\mathcal{G}^{\prime}$, and apply Lemma 6.3 to $B=A^{\prime\prime}_{k_{0}}$, $m=k_{0}$, $\mathcal{G}_{k_{0}}$, $\zeta=\zeta_{g}$ for an arbitrary $g\in\mathcal{G}^{\prime}$, and obtain $\mathcal{G}^{\prime}_{k_{0}}$, $A^{\prime\prime}_{k_{0}+1}$, $l_{k_{0}+1}$ and $\hat{\mathcal{B}}_{k_{0}+1,j}$ for $j<l_{k_{0}+1}$. We then repeat this step until for some $k_{1}\geq k_{0}$, $\mathcal{G}^{\prime}_{k_{1}}=\varnothing$. Such a $k_{1}$ exists since, by Lemma 6.3, $|\mathcal{G}^{\prime}_{m}|<|\mathcal{G}^{\prime}_{m-1}|$ for each $m>k_{0}$.

It follows from items (12) and (13) that $\bigcup_{i<k_{0}}\bigcup_{j<l_{i}}\hat{\mathcal{B}}_{i,j}$ and $\mathcal{G}\cup\{\chi_{\vec{\nu_{i}}}:i<k_{0}\}$, mod $p$, are bases for the same subspace of $G^{\omega}/p$. Let $\mathcal{A}=\mathcal{G}\cup\{\chi_{\vec{\nu_{i}}}:i<k_{0}\}$ and $\mathcal{B}^{\prime}=\bigcup_{i<k_{0}}\bigcup_{j<l_{i}}\hat{\mathcal{B}}_{i,j}$.

Fix families of integers $\hat{\mathcal{M}}=(\hat{\mathcal{M}}_{f,h}:f\in\mathcal{A},g\in\mathcal{B}^{\prime})$ and $\hat{\mathcal{N}}=(\hat{\mathcal{N}}_{h,f}:h\in\mathcal{B}^{\prime},f\in\mathcal{A})$ and a positive integer $T$ such that:

$(3^{\prime})$ $[f]_{p}=\frac{1}{T}\sum_{h\in{\mathcal{B}}^{\prime}}\hat{\mathcal{M}}_{f,h}[h]_{p}$, for each $f\in{\mathcal{A}}$ and

$(4^{\prime})$ $[h]_{p}=\frac{1}{T}\sum_{f\in{\mathcal{A}}}\hat{\mathcal{N}}_{h,f}[f]_{p}$, for each $h\in{\mathcal{B}}^{\prime}$.

Let ${\mathcal{C}}=\{\frac{h}{T}:\,h\in{\mathcal{B}}^{\prime}\}$ and ${\mathcal{M}}$, ${\mathcal{N}}$ be such that ${\mathcal{M}}_{f,\frac{h}{T}}=\hat{\mathcal{M}}_{f,h}$ and ${\mathcal{N}}_{\frac{h}{T},f}=\hat{\mathcal{N}}_{h,f}$. Then:

$(3^{\prime\prime})$ $[f]_{p}=\sum_{h\in{\mathcal{B}}}{\mathcal{M}}_{f,h}[h]_{p}$, for each $f\in{\mathcal{A}}$ and

$(4^{\prime\prime})$ $[h]_{p}=\frac{1}{T^{2}}\sum_{f\in{\mathcal{A}}}{\mathcal{N}}_{h,f}[f]_{p}$, for each $h\in{\mathcal{C}}$.

Let $A\subset A^{\prime\prime}$, $A\in p$ be such that for every $n\in A$, $f\in{\mathcal{A}}$ and $h\in{\mathcal{C}}$, $f(n)=\sum_{h\in{\mathcal{C}}}{\mathcal{M}}_{f,h}h(n)$ and $h(n)=\frac{1}{T^{2}}.\sum_{f\in{\mathcal{A}}}{\mathcal{N}}_{h,f}f(n)$.

Now let $K$ be a strictly increasing sequence of positive integers such that $K_{0}>1$, $n!T\mid K_{n}$ for all $n\in\omega$, and $K.\mathcal{C}\subset H^{\omega}$. We now have that by defining $\mathcal{B}_{i,j}=K.\frac{\hat{\mathcal{B}}_{i,j}}{T}$, we have the desired rational stack. ∎

7. Solving arc functions on a level of a rational stack

The main tool to solve the arc equation of rational stacks is the same used for integer stacks and we state the lemmas used in [8]. However, there is a crucial difference in the way the stack was defined so that we separate when the denominator grows compared to the numerator, when the numerator and denominator are pretty much at even speed and when the numerator grows compared to the denominator.

7.1. An application of Kronecker’s Lemma

Kronecker’s Lemma says that if $\{1,\theta_{0},\ldots,\theta_{k-1}\}$ is a linearly independent subset of the vector space ${\mathbb{R}}$ over the field ${\mathbb{Q}}$ then $\{(\theta_{0}.n+{\mathbb{Z}},\ldots,\theta_{k-1}.n+{\mathbb{Z}}):\,n\in{\mathbb{Z}}\}$ is a dense subset of ${\mathbb{T}}^{k}$ (see [1]).

Lemma 7.1**.**

( Lemma 4.3 [8]) If $(\theta_{0},\ldots,\theta_{r-1})$ is linearly independent subset of the $\mathbb{Q}$-vector space ${\mathbb{R}}$ and $\epsilon>0$ then there exists a positive integer $L$ such that $\{(\theta_{0}.x+{\mathbb{Z}},\ldots,\theta_{r-1}.x+{\mathbb{Z}}):\,x\in I\}$ is $\epsilon$-dense in the usual Euclidean metric topology, for any interval $I$ of length at least $L$.

Given $\epsilon>0$ and $\theta=(\theta_{i}:i\in I)$ a finite linearly independent subset of $\mathbb{R}$ as a $\mathbb{Q}$-vector space, fix an integer $L(\theta,\epsilon)$ satisfying the conditions in Lemma 7.1.

Lemma 7.2**.**

(Lemma 4.4 of [8]) Fix a positive real $\epsilon^{*}<\frac{1}{8}$. Let $\theta=(\theta_{0},\ldots,\theta_{r-1})$ be a linearly independent subset of ${\mathbb{R}}$ as a ${\mathbb{Q}}$-vector space.

Set $L=L(\theta,\epsilon^{*})$ and $(a_{0},\ldots,a_{r-1})$ be a sequence of integers such that

$i)$ $|a_{0}|>\ldots>|a_{r-1}|$ and

$ii)$ $|\theta_{k}-\frac{a_{k}}{a_{0}}|<\frac{\epsilon^{*}}{\sqrt{r}.L}$ for each $k<r$.

Then

$a)$ $\{(a_{0}.x,\ldots,a_{r-1}.x):\,x\in J\}$ is $2.\epsilon^{*}$-dense for any arc $J$ of length at least $\frac{L}{|a_{0}|}$ and

$b)$ for any arc $J$ of length at least $3.\frac{L}{|a_{0}|}$ and ${\mathcal{U}}$ any open ball of radius $4.\epsilon^{*}$ (in $\mathbb{T}^{r}$ with the Euclidean metric), there exists an arc $K$ contained in $J$ of length $\frac{4.\epsilon^{*}}{\sqrt{r}.|a_{0}|}$ such that $\{(a_{0}.x,\ldots,a_{r-1}.x):\,x\in K\}\subseteq{\mathcal{U}}$.

7.2. Solving arc equations on a level of a rational stack

We will present the proof of the remaining technical lemma.

Lemma 3.2.

Let $\mathcal{S}$, ${\mathcal{A}}$, ${\mathcal{C}}$, ${\mathcal{M}}$ and ${\mathcal{N}}$ be as in Lemma 3.1. Let $\epsilon$ be a positive real and $D$ be a finite subset of $\kappa$. Then there exist $B\subseteq A$ cofinite in $A$ and a family of positive real numbers $(\gamma_{n}:n\in B)$ such that:

For every $n\in B$, for every family $(W_{h}:\,h\in{\mathcal{C}})$ of open arcs of length $\epsilon$, and for every arc function $\psi$ of length $\epsilon$ such that $\operatorname{\text{supp }}\psi\subseteq D\setminus\{\nu_{0},\ldots\nu_{k_{0}-1}\}$, there exists an $n$-solution of length $\gamma_{n}$ for the arc equation $(\psi,B,K.\mathcal{C},K_{n},W)$.

Proof.

For each $(i,j)$ with $i<k_{1}$ and $j<l_{i}$, let $u_{i,j}\in\mathcal{B}_{i,j}$ and $v_{i,j}$ be such that for every $h\in\mathcal{B}_{i,j}$ and $n\in A$, $|u_{i,j}(n)(\zeta_{i}(n))|\leq|h(n)(\zeta_{i}(n))|\leq|v_{i,j}(n)(\zeta_{i}(n))|$. Fix $\epsilon^{*}<\min\{\frac{1}{8},\frac{1}{8}\epsilon\}$. For each $i<k_{1}$, $j<l_{i}$ and $h\in\mathcal{B}_{i,j}$, let $\theta_{h}$ be the limit of $\left(\frac{h(n)(\zeta_{i}(n))}{v_{i,j}(n)(\zeta_{i}(n))}\right)_{n\in A}$. Let $\theta_{i,j}=(\theta_{h}:h\in\mathcal{B}_{i,j})$ and let $L$ be a fixed integer greater than $L(\theta_{i,j},\epsilon^{*})$ for any $i<k_{1}$ and $j<l_{i}$.

Let $B\subset A$ be the set of $n$’s in $A$ is such that

$a)$ $\left|\theta_{h}-\frac{h(n)(\zeta_{i}(n))}{v_{i,j}(n)(\zeta_{i}(n))}\right|<\frac{\epsilon^{*}}{\sqrt{|{\mathcal{A}}|+1}.L}$, for each $i<k_{1}$, $j<l_{i}$ and $h\in\mathcal{B}_{i,j}$;

$b)$ $3.\frac{L}{|v_{i,l_{i}-1}(n)(\zeta_{i}(n))|}<\epsilon^{*}$, for each $i<k_{1}$.

$c)$ $\frac{3L}{|u_{i,j-1}(n)(\zeta_{i}(n))|}\leq\frac{4.\epsilon^{*}}{\sqrt{|{\mathcal{A}}|+1}.|v_{i,j}(n)(\zeta_{i}(n))|}$, for each $i<k_{1}$ and $j<l_{i}$, and

$d)$ $\{\zeta_{i}(n):\,k_{0}\leq i<k_{1}\}\cap D=\varnothing$.

Notice that $B$ is cofinite in $A$, therefore is in $p$. Let $\gamma_{n}=\frac{\epsilon^{*}}{(|\mathcal{A}|+1)\max\{\|h(n)\|:h\in\mathcal{B}\}}$ for each $n\in B$, where $\|h(n)\|=\sum_{\mu\in\operatorname{\text{supp }}h(n)}|h(n)(\mu)|$. Now let $(W_{h}:h\in\mathcal{C})$ and $\psi$ be given. Fix $n\in B$.

For each $h\in\mathcal{C}$, fix $V_{h}\subseteq W_{h}$ with $V_{h}$ an arc of length $4\epsilon^{*}$.

Given an arbitrary $\epsilon$-arc function $\psi$ as required, fix $\psi^{*}$ an $\epsilon^{*}$-arc function such that $\operatorname{\text{supp }}\psi^{*}\supseteq D$, $\operatorname{\text{supp }}\psi^{*}\cap\{\zeta_{i}(n):\,0\leq i<k_{1}\}=\varnothing$, $\psi^{*}\leq\psi$ and $\operatorname{\text{supp }}h(n)\setminus\{\zeta_{i}(n):\,k_{0}\leq i<k_{1}\}\subseteq\operatorname{\text{supp }}\psi^{*}$, for each $i<k_{1}$, $j<l_{i}$ and $h\in\mathcal{B}_{i,j}$.

For each $\mu\in\operatorname{\text{supp }}\psi^{*}$ choose $x_{\mu}$ such that $K_{n}x_{\mu}$ is the center of $\psi^{*}(\mu)$.

For each $0\leq i<k_{1}$, $j<l_{i}$, notice that $\{\zeta_{0}(n),\ldots,\zeta_{k_{1}-1}(n)\}\cap\operatorname{\text{supp }}h\subset\{\zeta_{i}(n),\ldots,\zeta_{k_{1}-1}(n)\}$ for each $h\in\mathbb{B}_{i,j}$.

We will define by downward recursion, for $0<i\leq k_{1}-1$, an arc $Q_{i,0}\subset\mathbb{T}$.

For the first step $i_{\star}=k_{1}-1$, we will define $Q_{i_{\star},j}$ for $j<l_{i_{\star}}$, also by downward recursion. So let $j_{\star}=l_{i_{\star}}-1$ be the first step.

Let $O_{h}=V_{h}-\sum_{\mu\operatorname{\text{supp }}h(n)\setminus\{\zeta_{i_{*}}(n)\}}h(n)(\mu)x_{\mu}$, for each $h\in\mathcal{B}_{i_{*},j*}$.

Fix an arbitrary arc $J$ of length at least $3.\frac{L}{|v_{i_{\star},j_{\star}}(n)(\zeta_{i_{\star}}(n))|}$ and ${\mathcal{U}}_{i_{\star},j_{\star}}$ the ball of radius $4.\epsilon^{*}$ contained in the product of $\prod_{h\in\mathcal{B}_{i_{\star},j_{\star}}}O_{h}$. By Lemma 7.2, there exists an interval $Q_{i_{\star},j_{\star}}$ contained in $J$ of length $\frac{4.\epsilon^{*}}{\sqrt{|{\mathcal{A}}|+1}.|v_{i_{\star},j_{\star}}(n)(\zeta_{i_{\star}}(n))|}$ such that $\{(h(n)(\zeta_{i_{\star}}(n)).x:h\in\mathcal{B}_{i_{\star},j_{\star}}):\,x\in Q_{i_{\star},j_{\star}}\}\subseteq{\mathcal{U}}_{i_{\star},j_{\star}}\subseteq\mathbb{T}^{\mathcal{B}_{i_{\star},j_{\star}}}$.

Suppose $j^{\prime}<l_{i_{\star}}$ and we have defined $Q_{i_{\star},j}$ for all $j^{\prime}\leq j<l_{i_{\star}}$. If $j^{\prime}=0$ then we are done for step $i_{\star}=k_{1}-1$.

If not: by $c)$, it follows that $\frac{3L}{|u_{i_{\star},j^{\prime}-1}(n)(\zeta_{i_{\star}}(n))|}\leq\frac{4.\epsilon^{*}}{\sqrt{|{\mathcal{A}}|+1}.|v_{i_{\star},j^{\prime}}(n)(\zeta_{i_{\star}}(n))|}$ and $Q_{i_{\star},j^{\prime}}$ has size exactly the right side of the inequality above. Let ${\mathcal{U}}_{i_{\star},j^{\prime}-1}$ be a ball of length $4\epsilon^{*}$ contained in $\prod_{h\in\mathcal{B}_{i_{\star},j^{\prime}-1}}O_{h}$. Applying Lemma 7.2, there exists an arc $Q_{i_{\star},j^{\prime}-1}$ of length $\frac{4.\epsilon^{*}}{\sqrt{|{\mathcal{A}}|+1}.|v_{i_{\star},j^{\prime}-1}(n)(\zeta_{i_{\star}}(n))|}$ contained in $Q_{i_{\star},j^{\prime}}$ such that $\{(h(n)(\zeta_{i_{\star}}(n)).x:h\in\mathcal{B}_{i_{\star},j^{\prime}-1}):\,x\in Q_{i_{\star},j^{\prime}-1}\}\subseteq{\mathcal{U}}_{i_{\star},j^{\prime}-1}$.

We thus obtain, for $i_{\star}=k_{1}-1$, $Q_{i_{\star},0}$ an arc of length $\frac{4.\epsilon^{*}}{\sqrt{|{\mathcal{A}}|+1}.|v_{i_{\star},0,}(n)(\zeta_{i_{\star}}(n))|}$ such that

$\{(h(n)(\zeta_{i_{\star}}).x:h\in\mathcal{B}_{i_{\star},j}):\,x\in Q_{i_{\star},j}\}\subseteq\prod_{h\in\mathcal{B}_{i_{\star},j}}O_{h}$, for each $j<l_{i_{\star}}$. At the end, we will have defined $Q_{i_{\star},0}$.

Let $x_{\zeta_{i_{\star}}(n)}$ be the center of $Q_{i_{\star},0}$. By definition of $O_{h}$ we have

$O_{h}=V_{h}-\sum_{\mu\in\operatorname{\text{supp }}h(n)\setminus\{\zeta_{i_{\star}}(n)\}}h(n)(\mu)x_{\mu}$, for each $j<l_{i_{\star}}$ and $h\in\mathcal{B}_{i_{\star},j}$.

Then $\sum_{\mu\in\operatorname{\text{supp }}h(n)}h(n)(\mu)x_{\mu}\in\sum_{\mu\in\operatorname{\text{supp }}h(n)\setminus\{\zeta_{i_{\star}}(n)\}}h(n)(\mu)x_{\mu}+O_{h}=V_{h}$, for each $j<l_{i_{\star}}$ and $h\in\mathcal{B}_{i_{\star},j}$.

The first step of the recursion has been carried out. Suppose $0\leq i^{\prime}<k_{1}$ and $Q_{i,0}$ has been defined for all $i^{\prime}\leq i<k_{1}$.

If $i^{\prime}=0$ then we are done. Otherwise if $i^{\prime}-1\geq 0$:

Let $O_{h}=V_{h}-\sum_{\mu\in\operatorname{\text{supp }}h(n)\setminus\{\zeta_{i^{\prime}-1}(n)\}}h(n)(\mu)x_{\mu}$, for each $j<l_{i^{\prime}-1}$ and $h\in\mathcal{B}_{i^{\prime}-1,j}$.

It is important here that $\{\zeta_{0}(n),\ldots,\zeta_{i^{\prime}-1}(n)\}\cap\ \operatorname{\text{supp }}h=\{\zeta_{i^{\prime}-1}(n)\}$, for each $j<l_{i^{\prime}-1}$ and $h\in\mathcal{B}_{i^{\prime}-1,j}$.

We are in similar conditions as for $i^{\prime}$ to obtain $Q_{i^{\prime}-1,0}$ and $x_{\zeta_{i^{\prime}-1(n)}}$ the center of $Q_{i^{\prime}-1,0}$ an arc of length $\frac{4.\epsilon^{*}}{\sqrt{|{\mathcal{A}}|+1}.|v_{i^{\prime}-1,0}(n)(\zeta_{i^{\prime}-1}(n))|}$ such that

$\sum_{\mu\in\operatorname{\text{supp }}h(n)}h(n)(\mu)x_{\mu}\in\sum_{\mu\in\operatorname{\text{supp }}h(n)\setminus\{\zeta_{i^{\prime}-1}(n)\}}h(n)(\mu)x_{\mu}+O_{h}=V_{h}$, for each $j<l_{i^{\prime}-1}$ and $h\in\mathcal{B}_{i^{\prime}-1,j}$.

This ends the construction of $x_{\mu}$ for each $\mu\in\operatorname{\text{supp }}\psi^{*}\cup\{\zeta_{0}(n),\ldots,\zeta_{k_{1}-1}(n)\}$. Choose an arbitrary $x_{\mu}$ for $\mu\in D\setminus(\operatorname{\text{supp }}\psi^{*}\cup\{\zeta_{0}(n),\ldots,\zeta_{k_{0}-1}(n)\})$.

Let $\phi(\mu)$ be the arc of center $x_{\mu}$ and length $\gamma_{n}$. We show that $\phi$ is the solution we are looking for.

By the choice of $x_{\mu}$ and since $\psi^{*}\leq\psi$, it follows $K_{n}.\phi\leq\psi$.

Secondly, if $h\in\mathcal{B}_{i,j}$ then $\sum_{\mu\in\operatorname{\text{supp }}h(n)}h(n)(\mu)x_{\mu}\in V_{h}$. It follows that the center of $\sum_{\mu\in\operatorname{\text{supp }}h(n)}h(n)(\mu)\phi(\mu)$ is contained in $V_{h}$ and this arc has length at most $\sum_{\mu\in\operatorname{\text{supp }}h(n)}|h(n)(\mu)|.\gamma_{n}=\sum_{\mu\in\operatorname{\text{supp }}h(n)}|h(n)(\mu)|.\frac{\epsilon^{*}}{(|\mathcal{A}|+1)\max\{\|h(n)\|:h\in\mathcal{B}\}}<\epsilon^{*}$. Therefore, a point of the arc $\sum_{\mu\in\operatorname{\text{supp }}h(n)}h(n)(\mu)\phi(\mu)$ is at a distance smaller than $2\epsilon^{*}+\epsilon^{*}$ from the center of $V_{h}$. Since $W_{h}$ and $V_{h}$ have the same center and $3\epsilon^{*}<\frac{\epsilon}{2}$ it follows that $\sum_{\mu\in\operatorname{\text{supp }}h(n)}h(n)(\mu)\phi(\mu)\subseteq W_{h}$. Thus, $\phi$ is as required.

∎

Dedication

The third listed author would like to dedicate this work to his long term collaborator and friend, Dr. Salvador Garcia Ferreira on the occasion of his 60th birthday.