Combined Properties of Finite Sums And Finite products near zero

TL;DR

This paper extends combinatorial properties of finite sums and products to the near-zero region in dense subsemigroups of (0,1), using advanced topological tools like the Stone-Čech compactification.

Contribution

It proves that properties of finite sums and products near zero hold in dense subsemigroups of (0,1), generalizing previous results on partitions and progressions.

Findings

Finite sums and products properties hold near zero in dense subsemigroups.

Partition results for progressions extend to the near-zero setting.

Uses Stone-Čech compactification to establish these properties.

Abstract

It was proved that whenever N is partitioned into finitely many cells, one cell must contain arbitrary length geo-arithmetic progressions. It was also proved that arithmetic and geometric progressions can be nicely intertwined in one cell of partition, whenever N is partitioned into finitely many cells. In this article we shall prove that similar types of results also hold near zero in some suitable dense subsemigroup S of ((0; 1) ; +), using the Stone- Cech compactification of ?S.

Equations

$$\left\{r^j\left(a+id\right):i,j\in \left\{1,2,\dots ,k\right\}\right\}.$$

$$\left\{b{\left(a+id\right)}^j:i,j\in \left\{0,1,\dots ,k\right\}\right\}\bigcup \left\{b{d}^j:j\in \left\{0,1,\dots ,k\right\}\right\}$$

$$\bigcup \left\{a+id:i\in \left\{0,1,\dots ,k\right\}\right\}$$

$$FS(⟨y_n{⟩}_{n=1}^{\infty })\cup FP(⟨y_n{⟩}_{n=1}^{\infty })\subseteq A.$$

$$\bigcap _{m=1}^{\infty }\overline{FS(⟨x_n{⟩}_{n=m}^{\infty })}\cap K(\beta \mathbb{N})\ne \varnothing .$$

$$FS(⟨x_n{⟩}_{n=1}^{\infty })\cup FP(⟨x_n{⟩}_{n=1}^{\infty })\subseteq A.$$

$$FS(⟨x_n{⟩}_{n=1}^{\infty })\cup FP(⟨x_n{⟩}_{n=1}^{\infty })\subseteq A_i.$$

$$FS(⟨x_n{⟩}_{n=1}^{\infty })\cup FP(⟨x_n{⟩}_{n=1}^{\infty })\subseteq A_i?$$

$$FS(⟨y_n{⟩}_{n=1}^{\infty })\cup FP(⟨y_n{⟩}_{n=1}^{\infty })\subseteq A.$$

$$D=B\cap A^{\star }\cap \bigcap _{s\in E_1}(-s+A^{\star })\cap \bigcap _{s\in E_2}(s^{-1}{A}^{\star })\cap \bigcap _{s\in E_2}(A^{\star }{s}^{-1}).$$

$$\left(\begin{array}[]{cc}M&O\\ O&N\end{array}\right)\,$$

Full text

1. Introduction

One of the famous Ramsey theoretic results is van der Waerden’s Theorem [11], which says that whenever the set $\mathbb{N}$ of natural numbers is divided into finitely many classes, one of these classes contains arbitrarily long arithmetic progressions. The analogous statement about geometric progressions is easily seen to be equivalent via the homomorphisms $p\colon\,\left(\mathbb{N},\,+\right)\to\left(\mathbb{N},\,\cdotp\right)$ such that $p\left(x\right)=2^{x}$, and $q\colon\,\left(\mathbb{N}\setminus\left\{1\right\},\,\cdotp\right)\to\left(\mathbb{N},\,+\right)$, where $q\left(x\right)$ is the length of the prime factorization of $x$.

It has been shown in [1, Theorem 3.11] that any set which is multiplicatively large, that is a piecewise syndetic IP set in $\left(\mathbb{N},\,\cdotp\right)$ must contain substantial combined additive and multiplicative structure; in particular it must contain arbitrarily large geo-arithmetic progressions, that is, sets of the form

[TABLE]

A well known extension of van der Waerden’s Theorem allows one to get the additive increment of the arithmetic progression in the same cell as the arithmetic progression. Similarly for any finite partition of $\mathbb{N}$ there exist some cell $A$ and $b,r\in\mathbb{N}$ such that $\left\{r,b,br,\ldots,br^{k}\right\}\subseteq A$. It is proved in [1, Theorem 1.5] that these two facts can be intertwined:

Theorem 1.1**.**

*Let $r,k\in\mathbb{N}$ and $\mathbb{N}=\bigcup_{i=1}^{r}A_{i}$. Then there exist $s\in\left\{1,2,\dots,r\right\}$and $a,b,d\in A_{s}$, such that

[TABLE]

We know that if $A\subseteq\mathbb{N}$ belongs to every idempotent in $\beta\mathbb{N}$, then it is called an $IP^{*}$ set. Given a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$, we let $FP(\langle x_{n}\rangle_{n=1}^{\infty})$ be the product analogue of Finite Sum. Given a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$, we say that $\langle y_{n}\rangle_{n=1}^{\infty}$ is a sum subsystem of $\langle x_{n}\rangle_{n=1}^{\infty}$ provided there is a sequence $\langle H_{n}\rangle_{n=1}^{\infty}$ of nonempty finite subsets of $\mathbb{N}$ such that $\max H_{n}<\min H_{n+1}$, and $y_{n}=\sum_{t\in H_{n}}\,x_{t}$ for each $n\in\mathbb{N}$.

Theorem 1.2**.**

Let $\langle x_{n}\rangle_{n=1}^{\infty}$ be a sequence in $\mathbb{N}$ and $A$ be an IP set in $(\mathbb{N},+)$. Then there exists a sum subsystem $\langle y_{n}\rangle_{n=1}^{\infty}$ of $\langle x_{n}\rangle_{n=1}^{\infty}$ such that*

[TABLE]

Proof.

[2, Theorem 2.6] or see [9, Corollary 16.21]. ∎

The algebraic structure of the smallest ideal of $\beta S$ has played a significant role in Ramsey Theory. It is known that any central subset of $(\mathbb{N},+)$ is guaranteed to have substantial additive structure. But Theorem 16.27 of [9] shows that central sets in $(\mathbb{N},+)$ need not have any multiplicative structure at all. On the other hand, in [2] we see that sets which belong to every minimal idempotent of N, called central* sets, must have significant multiplicative structure.

In case of central* sets a similar result has been proved in [4] for a restricted class of sequences called minimal sequences, where a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$ is said to be a minimal sequence if

[TABLE]

Theorem 1.3**.**

Let $\langle y_{n}\rangle_{n=1}^{\infty}$ be a minimal sequence and $A$ be a $central^{*}$ set in $(\mathbb{N},+)$. Then there exists a sum subsystem $\langle x_{n}\rangle_{n=1}^{\infty}$ of $\langle y_{n}\rangle_{n=1}^{\infty}$ such that

[TABLE]

Proof.

[2, Theorem 2.4]. ∎

A similar result in this direction in the case of dyadic rational numbers has been proved by Bergelson, Hindman and Leader.

Theorem 1.4**.**

There exists a finite partition $\mathbb{D}\setminus\{0\}=\bigcup_{i=1}^{r}A_{i}$ such that there do not exist $i\in\{1,2,\ldots,r\}$ and a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ with

[TABLE]

Proof.

[3, Theorem 5.9]. ∎

In [3], the authors also presented the following conjecture and question.

Conjecture 1.5**.**

There exists a finite partition $\mathbb{Q}\setminus\{0\}=\bigcup_{i=1}^{r}A_{i}$ such that there do not exists $i\in\{1,2,\ldots,r\}$ and a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ with

[TABLE]

Problem 1.6**.**

Does there exists a finite partition $\mathbb{R}\setminus\{0\}=\bigcup_{i=1}^{r}A_{i}$ such that there do not exists $i\in\{1,2,\ldots,r\}$ and a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ with

[TABLE]

In the section 2, we shall first work on some combined algebraic properties near [math] in the ring of quaternions, denoted by $\mathbb{H}$. The ring being non abelian, is a division ring having an idempotent [math]. In section 3, for any suitable dense subsemigroup $S$ of $\left(\left(0,\infty\right),+\right)$, our aim is to establish partitition regularity among two matrices using using additive and multiplicative structure of $\beta S$, Stone-Čech compactification of $S$.

2. Combined Algebraic and Multiplicative Properties near idempotent

in relation with Quaternion Rings

In the following discussion our aim is to extend Theorem 1.2 and Theorem 1.3 for dense subsemigroups $(\mathbb{H},+)$ in the appropriate context.

Definition 2.1**.**

If $S$ is a dense subsemigroup of $(\mathbb{H},+)$, we define $0^{+}(S)=\{p\in\beta S_{d}:(\forall r>0)(B_{d}(r)\in p)\}$.

It is proved in [7], that $0^{+}(S)$ is a compact right topological subsemigroup of $(\beta S_{d},+)$ which is disjoint from $K(\beta S_{d})$ and hence gives some new information which are not available from $K(\beta S_{d})$. Being compact right topological semigroup $0^{+}(S)$ contains minimal idempotents of $0^{+}(S)$. A subset $A$ of $S$ is said to be IP*-set near 0 if it belongs to every idempotent of $0^{+}(S)$ and a subset $C$ of $S$ is said to be central*∗* set near [math] if it belongs to every minimal idempotent of $0^{+}(S)$. In [5] the authors applied the algebraic structure of $0^{+}(S)$ on their investigation of image partition regularity near [math] of finite and infinite matrices. Article [6] used algebraic structure of $0^{+}(\mathbb{R})$ to investigate image partition regularity of matrices with real entries from $\mathbb{R}$.

Definition 2.2**.**

Let $S$ be a dense subsemigroup of $(\mathbb{H},+)$. A subset $A$ of $S$ is said to be an IP set near [math] if there exists a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ with $\sum_{n=1}^{\infty}x_{n}$ converges such that $FS(\langle x_{n}\rangle_{n=1}^{\infty})\subseteq A$. We call a subset $D$ of $S$ is an IP*∗* set near [math] if for every subset $C$ of $S$ which is IP set near [math], $C\cap D$ is IP set near [math].

From [10, Theorem 3.2], it follows that for a dense subsemigroup $S$ of $(\mathbb{H},+)$ a subset $A$ of $S$ is an IP set near [math] if only if there exists some idempotent $p\in 0^{+}(S)$ with $A\in p$. Further it can be easily observed that a subset $D$ of $S$ is an IP*∗* set near [math] if and only if it belongs to every idempotent of $0^{+}(S)$.

Given $c\in\mathbb{H}\setminus\{0\}$ and $p\in\beta\mathbb{H}_{d}$, the product $c\cdot p$ and $p\cdot c$ are defined in $(\beta\mathbb{H}_{d},\cdot)$. One has $A\subseteq\mathbb{H}$ is a member of $c\cdot p$ if and only if $c^{-1}A=\{x\in\mathbb{H}:c\cdot x\in A\}$ is a member of $p$ and similarly for $c\cdot p$.

Lemma 2.3**.**

Let $S$ be a dense subsemigroup of $(\mathbb{H},+)$ such that $S\cap\mathbb{H}$ is a subsemigroup of $(\mathbb{H}\setminus\{0\},\cdot)$. If $A$ is an IP set near [math] in $S$ then $sA$ is also an IP set near [math] for every $s\in S\cap B_{d}(1)\setminus\{0\}$. Further if $A$ is a an IP∗ set near [math] in $(S,+)$ then both $s^{-1}A$ and $As^{-1}$ are IP∗ set near [math] for every $s\in S\cap B_{d}(1)\setminus\{0\}$.

Proof.

Since $A$ is an IP set near [math] then by [7, Theorem 3.1] there exists a sequence $\langle x_{n}\rangle_{n=1}^{\infty}$ in $S$ with the property that $\sum_{n=1}^{\infty}x_{n}$ converges and FS$(\langle x_{n}\rangle_{n=1}^{\infty})\subseteq A$. This implies that $\sum_{n=1}^{\infty}(s\cdot x_{n})$ is also convergent and FS$(\langle sx_{n}\rangle_{n=1}^{\infty})\subseteq sA$. This proves that $sA$ is also an IP set near [math]. Similarly we can prove that $As^{-1}$ is also IP set near [math] for every $s\in S\cap B_{d}(1)\setminus\{0\}$. For the second let $A$ be a an IP*∗* set near [math] and $s\in S\cap B_{d}(1)\setminus\{0\}$. To prove that $s^{-1}A$ is a an IP*∗* set near [math] it is sufficient to show that if $B$ is any IP set near [math] then $B\cap s^{-1}A\neq\emptyset$. Since $B$ is an IP set near [math], $sB$ is also an IP set near [math] by the first part of the proof, so that $A\cap sB\neq\emptyset$. Choose $t\in sB\cap A$ and $k\in B$ such that $t=sk$. Therefore $k\in s^{-1}A$ so that $B\cap s^{-1}A\neq\emptyset$. ∎

Given $A\subseteq S$ and $s\in S$, $s^{-1}A=\{t\in S$ : $st\in A\}$ and $-s+A=\{t\in S$ : $s+t\in A\}$. In case of product we have to keep in mind the order of elements as the product is noncommutative here.

Definition 2.4**.**

Let $\langle x_{n}\rangle_{n=1}^{\infty}$ be a sequence in the ring $(\mathbb{H},+,\cdot)$, and let $k\in\mathbb{N}$. Then FP$(\langle x_{n}\rangle_{n=1}^{k})$ is the set of all products of terms of $\langle x_{n}\rangle_{n=1}^{k}$ in any order with no repetitions. Similarly FP$(\langle x_{n}\rangle_{n=1}^{\infty})$ is the set of all products of terms of $\langle x_{n}\rangle_{n=1}^{\infty}$ in any order with no repetitions.

Theorem 2.5**.**

Let $S$ be a dense subsemigroup of $(\mathbb{H},+)$, such that $S\cap B_{d}(1)\setminus\{0\}$ is a subsemigroup of $(B_{d}(1)\setminus\{0\},\cdot)$. Also let $\langle x_{n}\rangle_{n=1}^{\infty}$ be a sequence in $S$ such that $\sum_{n=1}^{\infty}x_{n}$ converges to [math] and $A$ be a IP∗ set near [math] in $S$. Then there exists a sum subsystem $\langle y_{n}\rangle_{n=1}^{\infty}$ of $\langle x_{n}\rangle_{n=1}^{\infty}$ such that

[TABLE]

Proof.

Since $\sum_{n=1}^{\infty}x_{n}$ converges to 0, from [7, Theorem 3.1] it follows that we can find some idempotent $p\in 0^{+}(S)$ for which $FS(\langle x_{n}\rangle_{n=1}^{\infty})\in p$. In fact $T=\bigcap_{m=1}^{\infty}\mbox{cl}_{\beta S_{d}}FS(\langle y_{n}\rangle_{n=m}^{\infty})\subseteq 0^{+}(S)$ and $p\in T$. Again, since $A$ is an IP* set near [math] in $S$, by the above Lemma 2.3 for every $s\in S\cap B_{d}(1)\setminus\{0\}$, both $s^{-1}A,As^{-1}\in p$. Let $A^{\star}=\{s\in A$ : $-s+A\in p\}$. Then by  [9, Lemma 4.14] $A^{\star}\in p$. We can choose $y_{1}\in A^{\star}\cap FS(\langle x_{n}\rangle_{n=1}^{\infty})$. Inductively let $m\in\mathbb{N}$ and $\langle y_{i}\rangle_{i=1}^{m}$, $\langle H_{i}\rangle_{i=1}^{m}$ in $\mathcal{P}_{f}(\mathbb{N})$ be chosen with the following properties:

1. (a)

$i\in\{1,2,\ldots,m-1\}$ $\max H_{i}<\min H_{i+1}$; 2. (b)

If $y_{i}=\sum_{t\in H_{i}}x_{t}$ then $\sum_{t\in H_{m}}x_{t}\in A^{\star}$ and $\mbox{FP}(\langle y_{i}\rangle_{i=1}^{m})\subseteq A$.

We observe that $\{\sum_{t\in H}x_{t}$ : $H\in\mathcal{P}_{f}(\mathbb{N}),\min H>\max H_{m}\}\in p$. Let $B=\{\sum_{t\in H}x_{t}$ : $H\in\mathcal{P}_{f}(\mathbb{N}),\min H>\max H_{m}\}$, let $E_{1}=FS(\langle y_{i}\rangle_{i=1}^{m})$ and $E_{2}=AP(\langle y_{i}\rangle_{i=1}^{m})$. Now consider

[TABLE]

Then $D\in p$. Now choose $y_{m+1}\in D$ and $H_{m+1}\in\mathcal{P}_{f}(\mathbb{N})$ such that $\min H_{m+1}>\max H_{m}$. Putting $y_{m+1}=\sum_{t\in H_{m+1}}x_{t}$ shows that the induction can be continued and this proves the theorem. ∎

3. An Application of additive and multiplicative structure of $\beta S$

We shall like to produce an alternative proof of the above Theorem 3.1 using additive and multiplicative structure of $\beta S$. We need the following notion.

Theorem 3.1**.**

Let $u,\,v\in\mathbb{N}$. Let $M$ be a finite image partition regular matrix over $\mathbb{N}$ of order $u\times v$, and let $N$ be an infinite image partition regular near [math] matrix over a dense subsemigroup $S$ of $((0,\infty),+)$. Then

[TABLE]

is image partition regular near [math] over $S$.

Definition 3.2**.**

Let $S$ be a subsemigroup of $((0,\infty),+)$ and let $A$ be a finite or infinite matrix with entries from $\mathbb{Q}$. Then $I(A)=\{p\in 0^{+}$ : for every $P\in p$, there exists $\vec{x}$ with entries from $S$ such that all entries of $A\vec{x}$ are in $P\}$.

The following lemma can be easily proved as [8, Lemma 2.5].

Lemma 3.3**.**

Let $A$ be a matrix, finite or infinite with entries from $\mathbb{Q}$.

(a) The set $I(A)$ is compact and $I(A)\neq\emptyset$ if and only if $A$ is image partition regular near [math].

(b) If $A$ is finite image partition regular matrix, then$I(A)$ is a sub-semigroup of $(0^{+},+)$.

Next, we shall investigate the multiplicative structure of $I(A)$. In the following Lemma 3.4, we shall see that if $A$ is a image partition regular near [math] then $I(A)$ is a left ideal of $(0^{+},\cdot)$. It is also a two-sided ideal of $(0^{+},\cdot)$, provided $A$ is a finite image partition regular near [math].

Lemma 3.4**.**

Let $A$ be a matrix, finite or infinite with entries from $\mathbb{Q}$.

(a) If $A$ is an image partition regular near [math], then $I(A)$ is a left ideal of $(0^{+},\cdot)$.

(b) If $A$ is a finite image partition regular near [math], then $I(A)$ is a two-sided ideal of $(0^{+},\cdot)$.

Proof.

(a). Let $A$ be a $u\times v$ image partition regular matrix, where $u,v\in\mathbb{N}\cup\{\omega\}$. Let $p\in 0^{+}$ and $q\in I(A)$. Also let $U\in p\cdot q$. Then

$\{x\in S:\,x^{-1}U\in q\}\in p$. Choose $z\in\{x\in S:\,x^{-1}U\in q\}$. Then $z^{-1}U\in q$. So there exists $\vec{x}$ with entries from $S$ such that $y_{j}\in z^{-1}U$ for $0\leq j<u$ where $\vec{y}=A\vec{x}$,

$\vec{y}=\left(\begin{matrix}y_{0}\\ y_{1}\\ y_{2}\\ \vdots\end{matrix}\right)$and $\vec{x}=\left(\begin{matrix}x_{0}\\ x_{1}\\ x_{2}\\ \vdots\end{matrix}\right)$; $\vec{y}$ and $\vec{x}$ are $u\times 1$ and $v\times 1$ matrices respectively. Now $y_{i}\in z^{-1}U$ for $0\leq i<u$ implies that $zy_{i}\in U$ for $0\leq i<u$. Let $\vec{x^{\prime}}=z\vec{x}$ and $\vec{y^{\prime}}=z\vec{y}$. Then $\vec{y^{\prime}}=A\vec{x^{\prime}}$. So there exists $\vec{x^{\prime}}$ with entries from $S$ such that all entries of $A\vec{x^{\prime}}$ are in $U$. Therefore $p.q\in I(A)$. So $I(A)$ is a left ideal of $(0^{+},\cdot)$.

(b) Let $A$ be a $u\times v$ matrix, where $u,v\in\mathbb{N}$. By previous lemma , $I(A)$ is a left ideal. We now show that $I(A)$ is a right ideal of $(0^{+},\cdot)$. Let $p\in\beta S$ and $q\in I(A)$. Now let $U\in q\cdot p$. Then $\{x\in S:x^{-1}U\in p\}\in q$. So there exists $\vec{x}$ with entries in $S$ such that $y_{i}\in\{x\in S:x^{-1}U\in p\}$ for $0\leq i<u$, where $\vec{y}=A\vec{x}$, $\vec{y}=\left(\begin{matrix}y_{0}\\ \vdots\\ y_{u-1}\end{matrix}\right)$ and $\vec{x}=\left(\begin{matrix}x_{0}\\ \vdots\\ x_{v-1}\end{matrix}\right)$; $\vec{y}$ and $\vec{x}$ are $u\times 1$ and $v\times 1$ matrices respectively. Now for $0\leq i<u,y_{i}\in\{x\in S:x^{-1}U\in p\}$. Hence $y_{i}^{-1}U\in p$ for $0\leq i<u$. This implies $\bigcap_{i=0}^{u-1}y_{i}^{-1}U\in p$. So $\bigcap_{i=0}^{u-1}y_{i}^{-1}U\neq\emptyset$. Let $z\in\bigcap_{i=0}^{u-1}y_{i}^{-1}U$. Therefore $z\in y_{i}^{-1}U$ for all $i\in\{0,1,2,.....,u-1\}$. Hence $y_{i}z\in U$ for $0\leq i<u$. Let $\vec{x^{\prime}}=\vec{x}z$ and $\vec{y^{\prime}}=\vec{y}z$. Then $\vec{y^{\prime}}=A\vec{x^{\prime}}$. So there exists $\vec{x^{\prime}}$ with entries from $S$ such that all entries of $A\vec{x^{\prime}}$ are in $U$. Thus $q\cdot p\in I(A)$. Therefore $I(A)$ is also a right ideal of $(\beta\mathbb{N},\cdot)$. Hence $I(A)$ is a two-sided ideal of $(0^{+},\cdot)$. ∎

Alternative proof of Theorem 2.2.7.

Let $r\in\mathbb{N}$ be given and $\epsilon>0$. Let $\mathbb{Q}=\bigcup_{i=1}^{r}E_{i}$. Suppose that $A$ be a $u\times v$ matrix where $u,v\in\mathbb{N}$. Also let $A=\left(\begin{matrix}M&0\\ 0&N\end{matrix}\right)$. Now by previous Lemma 3.4, $I(M)$ is a two-sided ideal of $(0^{+},\cdot)$. So $K(0^{+},\cdot)\subseteq I(A)$. Also by same Lemma 3.4, $I(N)$ is a left ideal of $(0^{+},\cdot)$. Therefore $K(0^{+},\cdot)\cap I(N)\neq\emptyset$. Hence $I(M)\cap I(N)\neq\emptyset$. Now choose $p\in I(M)\cap I(N)$. Since $\mathbb{Q}=\bigcup_{i=1}^{r}E_{i}$, there exists $k\in\{1,2,.....,r\}$ such that $E_{k}\in p$. Thus by definition of $I(M)$ and $I(N)$, there exist $\vec{x}\in S^{v}$ and $\vec{y}\in S^{\omega}$ such that $M\vec{x}\in E_{k}^{u}$ and $N\vec{y}\in E_{k}^{\omega}$. Take$\vec{z}=\left(\begin{matrix}\vec{x}\\ \vec{y}\end{matrix}\right)$. Then $A\vec{z}=\left(\begin{matrix}M\vec{x}\\ N\vec{y}\end{matrix}\right)$. So $A\vec{z}\in E_{k}^{\omega}$. Therefore $A=\left(\begin{matrix}M&0\\ 0&N\end{matrix}\right)$ is image partition regular near 0. ∎