Solvability of linear equations within weak mixing sets

TL;DR

This paper introduces WM sets, a new class of 'random' subsets of natural numbers, and establishes conditions for solving linear equations within these sets, including normal and partition-regular systems.

Contribution

It defines WM sets, explores their properties, and provides necessary and sufficient conditions for linear equation solvability within these sets, extending previous results to a broader class.

Findings

Solvability conditions for linear systems in WM sets

WM sets include normal sets as a subset

Partition-regular systems are solvable in any WM set

Abstract

We introduce a new class of "random" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that partition-regular system of linear equations with integer coefficients is solvable in any WM set.

Equations

$$2x_i=x_{i-1}+x_{i+1},i=2,3,\dots ,k-1.$$

$$z-y=x$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n=0}^{N-1}f(T^n\xi )={\int }_{X}f(x)d\mu (x).$$

$$X_{\xi }=\overline{\{T^n\xi {\}}_{n\in {\mathbb{N}}_0}}\subset \Omega .$$

$$\{n\vec{a}+m\vec{b}+\vec{f}|n,m\in \mathbb{N}\},$$

$$\det\left(\begin{array}[]{cc}a_{j}&b_{j}\\ c_{1,r}&c_{2,r}\\ \end{array}\right)\neq 0.$$

$$\forall r\in \{1,2,\dots ,l\}\exists c_r\in \mathbb{Z}suchthat\forall i\in F_r:f_i=c_r.$$

$$\{n\vec{a}+m\vec{b}+\vec{f}|n,m\in \mathbb{N}\},$$

$$\det\left(\begin{array}[]{cc}a_{j}&b_{j}\\ c_{1,r}&c_{2,r}\\ \end{array}\right)\neq 0.$$

$$S=\left\{n\in \mathbb{N}|n\alpha (\mathrm{mod}1)\in \left[\frac{1}{3},\frac{7}{12}\right]\right\}$$

$$<v,w{>}_N\doteq \frac{1}{N}\sum _{n=1}^{N}v(n)w(n).$$

$$\det\left(\begin{array}[]{cc}a_{i}&b_{i}\\ a_{j}&b_{j}\\ \end{array}\right)\neq 0.$$

$${\Vert w\Vert }_N<\epsilon ,$$

$$\left|\Vert v{\Vert }_N-d^k(A)\right|<\delta ,$$

$$v(n)=\frac{1}{M}\sum _{m=1}^M({\xi }_1(a_{1}n+b_{1}m)+d(A))\dots ({\xi }_k(a_{k}n+b_{k}m)+d(A)),$$

$$v(n)\doteq \frac{1}{M}\sum _{m=1}^M{1}_A(a_{1}n+b_{1}m+f_1)1_A(a_{2}n+b_{2}m+f_2)\dots 1_A(a_{l}n+b_{l}m+f_l)$$

$$(a_{1}n+b_{1}m+f_1,a_{2}n+b_{2}m+f_2,\dots ,a_{l}n+b_{l}m+f_l)\in A^l.$$

$$S_r=\{f_i-c_r|i\in F_r,f_i>c_r\}.$$

$$A_r=\{n\in \mathbb{N}\cup \{0\}|n\in Aandn+s\in A,\forall s\in S_r\}.$$

$$\frac{1}{N}\sum _{n=1}^N\frac{1}{M}\sum _{m=1}^M{1}_{A_1}(a_{1}n+b_{1}m)1_A(a_{2}n+b_{2}m)\dots 1_A(a_{l}n+b_{l}m)\approx \prod _{r=1}^{l}d(A_r).$$

$$(a_{1}n+b_{1}m+f_1,\dots ,a_{k}n+b_{k}m+f_k)\in A^k.$$

$${\psi }_{r,\vec{i}}^j(n)=\prod _{w\in \{0,1{\}}^j}\xi (n+r+w\cdot \vec{i}),n\in \mathbb{N},$$

$${\psi }_{r,\vec{i}}^j(n)=0,n\le 0.$$

$$\frac{1}{L}\sum _{l=1}^L{\left(\frac{1}{N}\sum _{n=1}^N{\psi }_{r,(c_1{i}_1,\dots ,c_j{i}_j)}^j(l+bn)\right)}^2<\epsilon$$

$$g_{r,\vec{i}}\doteq \prod _{ϵ\in V_j}{T}^{r+ϵ\cdot \vec{i}}\circ f,$$

$$g_{r,\vec{i}}^{\ast }=\prod _{ϵ\in V_j^{\ast }}{T}^{r+ϵ\cdot \vec{i}}\circ f,$$

$$g_{r,\vec{i}}(T^n\xi )={\psi }_{r,\vec{i}}^j(n).$$

$$\prod _{i=1}^k\frac{1}{N_i-M_i}\sum _{n\in [M_1,N_1)\times \dots [M_k,N_k)}\prod _{ϵ\in V_k^{\ast }}{T}^{{ϵ}_1{n}_1+\dots {ϵ}_k{n}_k}\circ f_{ϵ}$$

$$\prod _{ϵ\in V_k^{\ast }}{\int }_X{f}_{ϵ}d\mu$$

$${\int }_{X_{\xi }\times X_{\xi }}\prod _{i=1}^j\frac{1}{N_i-M_i}\sum _{\vec{i}\in [M_1,N_1)\times \dots \times [M_j,N_j)}{g}_{r,\vec{i}}(x)g_{r,\vec{i}}(y)d\mu (x)d\mu (y)=$$

Full text

1 Introduction

1.1 Algebraic patterns within subsets of $\mathbb{N}$

We use extensively the notion of “algebraic pattern”. By an algebraic pattern we mean a solution of a diophantine system of equations. For example, an arithmetic progression of length $k$ is an algebraic pattern corresponding to the following diophantine system:

[TABLE]

We investigate the problem of finding linear algebraic patterns (these correspond to linear systems) within a family of subsets of natural numbers satisfying some asymptotic conditions.

For instance, by Szemerédi theorem, subsets of positive upper Banach density (all $S\subset\mathbb{N}:d^{*}(S)>0$, where $d^{*}(S)=\limsup_{b_{n}-a_{n}\to\infty}\frac{|S\cap[a_{n},b_{n}]|}{b_{n}-a_{n}+1}$) contain the pattern of an arithmetic progression of any finite length (see [12]).

On the other hand, Schur patterns, namely triples of the form $\{x,y,x+y\}$, which correspond to solutions of the so-called Schur equation, $x+y=z$, do not necessarily occur in sets of positive upper density. For example, the odd numbers do not contain this pattern. But if we take a random subset of $\mathbb{N}$ by picking natural numbers with probability $\frac{1}{2}$ independently, then this set contains the Schur pattern with probability $1$.

There is a deterministically defined analog of a random set - a normal set. To define a normal set we recall the notions of a normal infinite binary sequence and of a normal number.

An infinite $\{0,1\}$-valued sequence $\lambda$ is called a normal sequence if every finite binary word $w$ occurs in $\lambda$ with frequency $\frac{1}{2^{|w|}}$, where $|w|$ is the length of $w$.

The more familiar notion is that of a normal number $x\in[0,1]$. If to a number $x\in[0,1]$ we associate its dyadic expansion $x=\sum_{i=1}^{\infty}\frac{x_{i}}{2^{i}}\,$ with $x_{i}\in\{0,1\}$, then $x$ is called a normal number if the sequence $(x_{1},x_{2},\ldots,x_{n},\ldots)$ is a normal sequence.

Definition 1.1.1

A set $S\subset\mathbb{N}$ is called normal if the 0-1 sequence $1_{S}$ ($1_{S}(n)=1\Leftrightarrow n\in S$) is normal.

Normal sets exhibit a non-periodic, “random” behavior. We notice that if $S$ is a normal set then $S-S$ contains $\mathbb{N}$. Therefore, the equation

[TABLE]

is solvable within every normal set. This implies that every normal set contains Schur patterns.

Normal sets are related to a class of dynamical systems displaying maximal randomness; namely Bernoulli systems. In this work we investigate occurrence of linear patterns in sets corresponding to dynamical systems with a lower degree of randomness, so called weakly mixing dynamical systems. The sets we obtain will be called WM sets. We will make this precise in the next section.

In the present paper we treat the following problem:

Give a complete characterization of the linear algebraic patterns which occur in all WM sets.

Remark 1.1.1

It will follow from our definition of a WM set, that any normal set is a WM set.

The problem of the solvability of a nonlinear equation or system of equations is beyond the limits of the technique used in this paper. Nevertheless, some particular equations might be analyzed. In [3] it is shown that there exist normal sets in which the multiplicative Schur equation $xy=z$ is not solvable.

1.2 Generic points and WM sets

For a formal definition of WM sets we need the notions of measure preserving systems and of generic points.

Definition 1.2.1

Let $X$ be a compact metric space, $\mathbb{B}$ the Borel $\sigma$-algebra on $X$; let $T:X\rightarrow X$ be a continuous map and $\mu$ a probability measure on $\mathbb{B}$. The quadruple $(X,\mathbb{B},\mu,T)$ is called a measure preserving system if for every $B\in\mathbb{B}$ we have $\mu(T^{-1}B)=\mu(B)$.

For a compact metric space $X$ we denote by $C(X)$ the space of continuous functions on $X$ with the uniform norm.

Definition 1.2.2

Let $(X,\mathbb{B},\mu,T)$ be a measure preserving system. A point $\xi\in X$ is called generic for the system $(X,\mathbb{B},\mu,T)$ if for any $f\in C(X)$ we have

[TABLE]

Example: Consider the Bernoulli system: $(X=\{0,1\}^{\mathbb{N}_{0}},\mathbb{B},\mu,T)$, where $X$ is endowed with the Tychonoff topology, $\mathbb{B}$ is Borel $\sigma$-algebra on $X$, $T$ is the shift to the left, $\mu$ is the product measure of $\mu_{i}$’s where $\mu_{i}(0)=\mu_{i}(1)=\frac{1}{2}$ and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$. An alternative definition of a normal set which is purely dynamical is the following.

A set $S$ is normal if and only if the sequence $1_{S}\in\{0,1\}^{\mathbb{N}_{0}}$ is a generic point of the foregoing Bernoulli $\left(\frac{1}{2},\frac{1}{2}\right)$ system.

The notion of a WM set generalizes that of a normal set, where the role played by Bernoulli dynamical system is taken over by dynamical systems of more general character.

Let $\xi(n)$ be any $\{0,1\}-$valued sequence. There is a natural dynamical system $(X_{\xi},T)$ connected to the sequence $\xi$:

On the compact space $\Omega=\{0,1\}^{\mathbb{N}_{0}}$ endowed with the Tychonoff topology, we define a continuous map $T:\Omega\longrightarrow\Omega$ by $(T\omega)_{n}=\omega_{n+1}$. Now for any $\xi$ in $\Omega$ we define

[TABLE]

Let $A$ be a subset of $\mathbb{N}$. Choose $\xi=1_{A}$ and assume that for an appropriate measure $\mu$, the point $\xi$ is generic for $(X_{\xi},\mathbb{B},\mu,T)$. We can attach to the set $A$ dynamical properties associated with the system $(X_{\xi},\mathbb{B},\mu,T)$.

We recall the notions of ergodicity, total ergodicity and weak-mixing in ergodic theory:

Definition 1.2.3

*A measure preserving system $(X,\mathbb{B},\mu,T)$ is called ergodic if every $A\in\mathbb{B}$ which is invariant under $T$, i.e. $T^{-1}(A)=A$, satisfies $\mu(A)=0$ or $1$.

A measure preserving system $(X,\mathbb{B},\mu,T)$ is called totally ergodic if for every $n\in\mathbb{N}$ the system $(X,\mathbb{B},\mu,T^{n})$ is ergodic.

A measure preserving system $(X,\mathbb{B},\mu,T)$ is called weakly mixing if the system $(X\times X,\mathbb{B}_{X\times X},\mu\times\mu,T\times T)$ is ergodic.*

In our discussion of WM sets corresponding to weakly mixing systems, we shall add the proviso that the dynamical system in question not be the trivial $1$-point system supported on the point $x\equiv 0$. This implies that the “density” of the set in question be positive.

Definition 1.2.4

Let $S\subset\mathbb{N}$. If the limit of $\frac{1}{N}\sum_{n=1}^{N}1_{S}(n)$ exists as $N\rightarrow\infty$ we call it the density of $S$ and denote by $d(S)$.

Definition 1.2.5

A subset $S\subset\mathbb{N}$ is called a WM set if $1_{S}$ is a generic point of the weakly mixing system $(X_{1_{S}},\mathbb{B},\mu,T)$ and $d(S)>0$.

1.3 Solvability of linear diophantine systems within WM

sets and normal sets

Our main result is a complete characterization of linear systems of diophantine equations which are solvable within every WM set. The characterization is given by describing affine subspaces of $\mathbb{Q}^{k}$ which intersect $A^{k}$, for any WM set $A\subset\mathbb{N}$.

Theorem 1.3.1

An affine subspace of $\mathbb{Q}^{k}$ intersects $A^{k}$ for every WM set $A\subset\mathbb{N}$ if and only if it contains a set of the form

[TABLE]

*where $\vec{a},\vec{b},\vec{f}$ have the following description:

$\vec{a}=(a_{1},a_{2},\ldots,a_{k})^{t}$, $\vec{b}=(b_{1},b_{2},\ldots,b_{k})^{t}\in\mathbb{N}^{k}$, $\vec{f}=(f_{1},f_{2},\ldots,f_{k})^{t}\in\mathbb{Z}^{k}$ and there exists a partition $F_{1},\ldots,F_{l}$ of $\{1,2,\ldots,k\}$ such that:

a$\mathrm{)}$ for every $r\in\{1,\ldots,l\}$ there exist $c_{1,r},c_{2,r}\in\mathbb{N}$, such that for every $i\in F_{r}$ we have $a_{i}=c_{1,r}\,,\,b_{i}=c_{2,r}$ and for every $j\in\{1,\ldots,k\}\setminus F_{r}$ we have*

[TABLE]

b$\mathrm{)}$

[TABLE]

We also classify all affine subspaces of $\mathbb{Q}^{k}$ which intersect $A^{k}$ for any normal set $A\subset\mathbb{N}$.

Theorem 1.3.2

An affine subspace of $\mathbb{Q}^{k}$ intersects $A^{k}$ for every normal set $A\subset\mathbb{N}$ if and only if it contains a set of the form

[TABLE]

*where $\vec{a},\vec{b},\vec{f}$ have the following description:

$\vec{a}=(a_{1},a_{2},\ldots,a_{k})^{t}$, $\vec{b}=(b_{1},b_{2},\ldots,b_{k})^{t}\in\mathbb{N}^{k}$, $\vec{f}=(f_{1},f_{2},\ldots,f_{k})^{t}\in\mathbb{Z}^{k}$ and there exists a partition $F_{1},\ldots,F_{l}$ of $\{1,2,\ldots,k\}$ such that for every $r\in\{1,\ldots,l\}$ there exist $c_{1,r},c_{2,r}\in\mathbb{N}$, such that for every $i\in F_{r}$ we have $a_{i}=c_{1,r}\,,\,b_{i}=c_{2,r}$ and for every $j\in\{1,\ldots,k\}\setminus F_{r}$ we have*

[TABLE]

A family of linear algebraic patterns that has been studied previously are the “partition regular” patterns. These are patterns which for any finite partition of $\mathbb{N}$: $\mathbb{N}=C_{1}\cup C_{2}\cup\ldots\cup C_{r}$, the pattern necessarily occurs in some $C_{j}$. (For example by van der Waerden’s theorem, arithmetic progressions are partition regular and by Schur’s theorem the Schur pattern is also partition regular). A theorem of Rado gives a complete characterization of such patterns. We will show in Proposition LABEL:Rado_analog that every linear algebraic pattern which is partition-regular occurs in every WM set.

It is important to mention that if we weaken the requirement of weak mixing to total ergodicity, then in the resulting family of sets, Rado’s patterns need not necessarily occur. For example, for $\alpha\not\in\mathbb{Q}$ the set

[TABLE]

is totally ergodic, i.e., $1_{S}$ is a generic point for a totally ergodic system and the density of $S$ is positive, but the equation $x+y=z$ is not solvable within $S$.

In the separate paper [4] we will address the question of solvability of more general algebraic patterns, not necessarily linear, in totally ergodic and WM sets.

The structure of the paper is the following. In Section LABEL:lin_proof_suff we prove the direction “$\Leftarrow$” of Theorems LABEL:main_thm_lin and LABEL:normal_theorem. In Section LABEL:lin_useful_constructions, by use of a probabilistic method, we prove the direction “$\Rightarrow$” of Theorems LABEL:main_thm_lin and LABEL:normal_theorem. In Section LABEL:sub_sect_Rado we show that every linear system which is solvable in one of the cells of any finite partition of $\mathbb{N}$ is also solvable within every WM set. The paper ends with Appendix in which we collected proofs of technical statements which have been used in Sections LABEL:lin_proof_suff and LABEL:lin_useful_constructions.

1.4 Acknowledgments

This paper is a part of the author’s Ph.D. thesis. I thank my advisor Prof. Hillel Furstenberg for introducing me to ergodic theory and for many useful ideas which I learned from him. I thank Prof. Vitaly Bergelson for fruitful discussions and valuable suggestions. Also, I would like to thank an anonymous referee for numerous valuable remarks.

2 Proof of Sufficiency

Notation: We introduce the scalar product of two vectors $v,w$ of length $N$ as follows:

[TABLE]

*We denote by $L^{2}(N)$ the (finite-dimensional) Hilbert space of all real vectors of length $N$ with the aforementioned scalar product.

We define: $\parallel{w}\parallel_{N}^{2}\doteq<w,w>_{N}$.*

First we state the following proposition which will prove useful in the proof of the sufficiency of the conditions of Theorem LABEL:main_thm_lin.

Proposition 2.1

*Let $A_{i}\subset\mathbb{N}$ \rm($$1\leq i\leq k$$\rm) be WM sets. Let

$\xi_{i}(n)\doteq 1_{A_{i}}(n)-\mathrm{d}(A_{i})$, where $\mathrm{d}(A_{i})$ denotes density of $A_{i}$. Suppose there are $(a_{1},b_{1}),(a_{2},b_{2}),\ldots,(a_{k},b_{k})\in\mathbb{Z}^{2}$, such that $a_{i}>0,\,1\leq i\leq k$, and for every $i\neq j$*

[TABLE]

Then for every $\varepsilon>0$ there exists $M(\varepsilon)\in\mathbb{N}$, such that for every $M\geq M(\varepsilon)$ there exists $N(M,\varepsilon)\in\mathbb{N}$, such that for every $N\geq N(M,\varepsilon)$

[TABLE]

where $w(n)\doteq\frac{1}{M}\sum_{m=1}^{M}\xi_{1}(a_{1}n+b_{1}m)\xi_{2}(a_{2}n+b_{2}m)\ldots\xi_{k}(a_{k}n+b_{k}m)$ for every $n=1,2,\ldots,N$.

Since the proof of Proposition LABEL:main_prop involves many technical details, first we show how our main result follows from it. Afterwards we state and prove all the lemmas necessary for the proof of Proposition LABEL:main_prop.

We use an easy consequence of Proposition LABEL:main_prop.

Corollary 2.1

*Let $A$ be a WM set. Let $k\in\mathbb{N}$, suppose

$(a_{1},b_{1}),(a_{2},b_{2}),\ldots,(a_{k},b_{k})\in\mathbb{Z}^{2}$ satisfy all requirements of Proposition LABEL:main_prop and suppose $f_{1},\ldots,f_{k}\in\mathbb{Z}$. Then for every $\delta>0$ there exists $M(\delta)$ such that $\forall\,M\geq M(\delta)$ there exists $N(M,\delta)$ such that $\forall\,N\geq N(M,\delta)$ we have*

[TABLE]

where $v(n)\doteq\frac{1}{M}\sum_{m=1}^{M}1_{A}(a_{1}n+b_{1}m+f_{1})1_{A}(a_{2}n+b_{2}m+f_{2})\ldots 1_{A}(a_{k}n+b_{k}m+f_{k})$ for every $n=1,2,\ldots,N$.

**Proof. ** We rewrite $v(n)$ in the following form:

[TABLE]

for every $n=1,2,\ldots,N$. We introduce normalized WM sequences $\xi_{i}(n)=\xi(n+f_{i})$ (of zero average), where $\xi(n)=1_{A}(n)-d(A)$. By use of triangular inequality and Proposition LABEL:main_prop it follows that for big enough $M$ and $N$ (which depends on $M$) $\|v\|_{N}$ is as close as we wish to $d^{k}(A)$. This finishes the proof.

$\Box$

**Proof. ** (of Theorem LABEL:main_thm_lin, $\Lleftarrow$) Let $A\subset\mathbb{N}$ be a WM set. Without loss of generality, we can assume that for every $r:1\leq r\leq l$ we have $r\in F_{r}$.

It follows from Corollary LABEL:corol_main_prop that the vector $v$ defined by

[TABLE]

for every $n=1,2,\ldots,N$, is not identically zero for big enough $M$ and $N$. But this is possible only if for some $n,m\in\mathbb{N}$ we have

[TABLE]

The latter implies that $A^{k}$ intersects the affine subspace.

$\Box$

**Proof. ** (of Theorem LABEL:normal_theorem, $\Lleftarrow$) For every $r:\,1\leq r\leq l$ take all indices which comprise $F_{r}$. Denote this sequence of indices by $I_{r}$. Denote $c_{r}=\min_{i\in I_{r}}f_{i}$. Let $S_{r}$ be the set of all non-zero shifts of $f_{i},i\in F_{r},$ centered at $c_{r}$, i.e.,

[TABLE]

For example, if the sequence of $f_{i}$’s where $i\in F_{1}$ is $(-5,2,3,2,-5)$, then $S_{1}=\{7,8\}$.

Let $A$ be a normal set. For every $r\,:\,1\leq r\leq l$ we define sets $A_{r}$ by

[TABLE]

Then $A_{r}$ is no longer a normal set provided that $S_{r}\neq\emptyset$ ($d(A)=\frac{1}{2^{1+|S_{r}|}}$). But, for all $r\,:\,1\leq r\leq l$ the sets $A_{r}$’s are WM sets.

Without loss of generality, assume that for every $r\,:\,1\leq r\leq l$ we have $r\in F_{r}$.

From Proposition LABEL:main_prop it follows that for big enough $M$ and $N$

[TABLE]

The latter ensures that there exist $m,n\in\mathbb{N}$ such that

[TABLE]

$\Box$

Now we state and prove all the claims that are required in order to prove Proposition LABEL:main_prop.

Definition 2.1

Let $\xi$ be a WM-sequence ($\xi$ is a generic point for a weakly mixing system $(X_{\xi},\mathbb{B}_{X_{\xi}},\mu,T)$) of zero average. The autocorrelation function of $\xi$ of length $j\in\mathbb{N}$ with the shifts $\vec{i}=(i_{1},i_{2},\ldots,i_{j})\in\mathbb{Z}^{j}$ and $r\in\mathbb{Z}$ is the sequence $\psi_{r,\vec{i}}^{j}$ which is defined by

[TABLE]

where $w\cdot\vec{i}$ is the usual scalar product in $\mathbb{Q}^{j}$, and

[TABLE]

Lemma 2.1

Let $\xi$ be a WM-sequence of zero average and suppose $\varepsilon,\delta>0,\,b\in\mathbb{Z}\setminus\{0\}$. Then for every $j\geq 1$, $(c_{1},c_{2},\ldots,c_{j})\in(\mathbb{Z}\setminus\{0\})^{j}$ and $(r_{1},r_{2},\ldots,r_{j})\in\mathbb{Z}^{j}$ there exist $I=I(\varepsilon,\delta,c_{1},\ldots,c_{n})$, a set $S\subset[-I,I]^{j}$ of density at least $1-\delta$ and $N(S,\varepsilon)\in\mathbb{N}$, such that for every $N\geq N(S,\varepsilon)$ there exists $L(N,S,\varepsilon)$ such that for every $L\geq L(N,S,\varepsilon)$

[TABLE]

for every $(i_{1},i_{2},\ldots,i_{j})\in S$, where $r=\sum_{k=1}^{j}r_{k}$.

**Proof. ** We note that it is sufficient to prove the lemma in the case $c_{1}=c_{2}=\ldots=c_{j}=1$, since if the average of nonnegative numbers over a whole lattice is small, then the average over a sublattice of a fixed positive density is also small.

Recall that $\xi\in X_{\xi}\doteq\overline{\{T^{n}\xi\}_{n=0}^{\infty}}\subset supp({\xi})^{\mathbb{N}_{0}}$, where $T$ is the usual shift to the left on the dynamical system $supp({\xi})^{\mathbb{N}_{0}}$, and by the assumption that $\xi$ is a WM-sequence of zero average it follows that $\xi$ is a generic point of the weakly mixing system $(X_{\xi},\mathbb{B_{X_{\xi}}},\mu,T)$ and the function $f\,:\,f(\omega)\doteq\omega_{0}$ has zero integral.

Denote $\vec{i}=(i_{1},\ldots,i_{j})$.

We define functions $g_{r,\vec{i}},g^{*}_{r,\vec{i}}$ on $X_{\xi}$ by

[TABLE]

[TABLE]

where $V_{j}$ is the $j$-dimensional discrete cube $\{0,1\}^{j}$ and $V_{j}^{*}$ is the $j$-dimensional discrete cube except the zero point.

Notice that

[TABLE]

We use the following theorem which is a special case of a multiparameter weakly mixing PET of Bergelson and McCutcheon (theorem A.1 in [2]; it is also a corollary of Theorem 13.1 of Host and Kra in [9]).

• Let $(X,\mu,T)$ be a weakly mixing system. Given an integer $k$ and $2^{k}$ bounded functions $f_{\epsilon}$ on $X$, $\epsilon\in V_{k}$ , the functions*

[TABLE]

converge in $L^{2}(\mu)$ to the constant limit

[TABLE]

*when $N_{1}-M_{1},\ldots,N_{k}-M_{k}$ tend to $+\infty$. *

From this theorem applied to the weakly mixing system $X_{\xi}\times X_{\xi}$ and the functions $f_{\epsilon}(x)=T^{r}\circ f\otimes T^{r}\circ f$ for every $\epsilon\in V_{j}$, we obtain for every Folner sequence $\{F_{n}\}$ in $\mathbb{N}^{j}$ that an average over the multi-index $\vec{i}=\{i_{1},\ldots,i_{j}\}$ of $g^{*}_{r,\vec{i}}\otimes g^{*}_{r,\vec{i}}$ on $F_{n}$’s converges to zero in $L^{2}(\mu)$ (the integral of $T^{r}\circ f\otimes T^{r}\circ f$ is zero). Thus

[TABLE]

[TABLE]

as $N_{1}-M_{1},\ldots,N_{j}-M_{j}\to\infty$.

As a result we obtain the following statement:

For every $\varepsilon>0$, $j\in\mathbb{N}$ and every fixed $(r_{1},r_{2},\ldots,r_{j})\in\mathbb{N}^{j}$, there exists a subset $R\subset\mathbb{N}^{j}$ of lower density equal to one, such that

[TABLE]

for every $\vec{i}\in R$, where $r=\sum_{k=1}^{j}r_{j}$.

Recall that lower density of a subset $R\subset\mathbb{N}^{j}$ is defined to be

[TABLE]

Recall that $\psi^{j}_{r,\vec{i}}(l+bn)=g_{r,\vec{i}}\left(T^{l+bn}\xi\right)$.

The definition of the sequences $\psi^{j}$ implies

[TABLE]

[TABLE]

for any $r_{1},r_{2}\in\mathbb{Z}$, where $\vec{i}=(i_{1},\ldots,i_{j})$.

Therefore, in order to prove Lemma LABEL:autocorrelation1  it is sufficient to show the following:

For every $\varepsilon,\delta>0$ and for any a priori chosen $b\in\mathbb{N}$ there exists $I(\varepsilon,\delta)\in\mathbb{N}$, such that for every $I\geq I(\varepsilon,\delta)$ there exists a subset $S\subset[1,I]^{j}$ of density at least $1-\delta$ $\mathrm{(}$namely, we have \frac{|S\cap[1,I)^{j}|}{I^{j}}\geq 1-\delta$$\mathrm{)} and $N(S,\varepsilon)\in\mathbb{N}$, such that for every $N\geq N(S,\varepsilon)$ there exists $L(N,S,\varepsilon)\in\mathbb{N}$ such that for every $L\geq L(N,S,\varepsilon)$ the following holds for every $\vec{i}\in S$:

[TABLE]

Let $b\in\mathbb{N}$. Continuity of the function

$g_{0,\vec{i}}$ and genericity of the point $\xi\in X_{\xi}$ yield

[TABLE]

[TABLE]

[TABLE]

By applying the von Neumann ergodic theorem to the ergodic system

$(X_{\xi},\mathbb{B},\mu,T^{b})$ (ergodicity follows from weak-mixing of the original measure preserving system $(X_{\xi},\mathbb{B},\mu,T)$) we have

[TABLE]

From (LABEL:eq:one_lemma311) there exists $I(\varepsilon,\delta)\in\mathbb{N}$ big enough that for every $I\geq I(\varepsilon,\delta)$ there exists a set $S\subset[1,I]^{j}$ of density at least $1-\delta$ such that

[TABLE]

for all $\vec{i}\in S$.

From equation (LABEL:imp_eq_22) it follows that there exists $N(S,\varepsilon)\in\mathbb{N}$, such that for every $N\geq N(S,\varepsilon)$ we have

[TABLE]

for all $\vec{i}\in S$.

Finally, equation (LABEL:eq_lemm_2) implies that there exists $L(N,S,\varepsilon)\in\mathbb{N}$, such that for every $L\geq L(N,S,\varepsilon)$ we have

[TABLE]

for all $\vec{i}\in S$.

$\Box$

The following lemma is a generalization of the previous lemma to a product of several autocorrelation functions.

Lemma 2.2

*Let $\psi^{1,j}_{r_{1},\vec{i}},\ldots,\psi^{k,j}_{r_{k},\vec{i}}$ be autocorrelation functions of length $j$ of WM-sequences $\xi_{1},\ldots,\xi_{k}$ of zero average,

$\{c_{1}^{1},\ldots,c_{j}^{1},\ldots,c_{1}^{k},\ldots,c_{j}^{k}\}\in(\mathbb{Z}\setminus\{0\})^{jk}$ and $\varepsilon,\delta>0$. Suppose

$(a_{1},b_{1}),(a_{2},b_{2}),\ldots,(a_{k},b_{k})\in\mathbb{Z}^{2}$, such that $a_{i}>0$ for all $i:\,1\leq i\leq k$ and for every $i\neq j$*

[TABLE]

(If $k=1$ assume that $b_{1}\neq 0$.)*

Then there exists $I(\varepsilon,\delta)\in\mathbb{N}$, such that for every $I\geq I(\varepsilon,\delta)$ there exist $S\subset[-I,I]^{j}$ of density at least $1-\delta$, $M(S,\varepsilon)\in\mathbb{N}$, such that for every $M\geq M(S,\varepsilon)$ there exists $X(M,S,\varepsilon)\in\mathbb{N}$, such that for every $X\geq X(M,S,\varepsilon)$*

[TABLE]

for every $(i_{1},i_{2},\ldots,i_{j})\in S$.

**Proof. ** The proof is by induction on $k$.

THE CASE $k=1$ (and arbitrary $j$):

If $a_{1}=1$ then the statement of the lemma follows from Lemma LABEL:autocorrelation1. If $a_{1}>1$ then by Proposition LABEL:wm_prop of Appendix for a given $\vec{i}=(i_{1},\ldots,i_{j})\in S$ we have

[TABLE]

[TABLE]

(Limits exist by genericity of the point $\xi$.)

By Lemma LABEL:autocorrelation1 the right hand side of (LABEL:sub_eq_1) is small for large enough $M$. So, for large enough $X$ (depending on $M$ and $(i_{1},\ldots,i_{j})$) the statement of the lemma is true. By finiteness of $S$ we conclude that the statement of the lemma holds for $k=1$.

GENERAL CASE ($k>1$):

Suppose that the statement holds for $k-1$.

Denote

[TABLE]

Let $\varepsilon,\delta>0$. We show that there exists $\mathbb{I}(\varepsilon,\delta)\in\mathbb{N}$ such that for every $\mathbb{I}>\mathbb{I}(\varepsilon,\delta)$ a set $S\subset[-\mathbb{I},\mathbb{I}]^{j}$ of density at least $1-\delta$ can be chosen satisfying the following property:

There exists $I(\varepsilon,S)\in\mathbb{N}$ such that for every $I>I(\varepsilon,S)$ there exists $M(I)\in\mathbb{N}$ such that for all $M>M(I)$ for a set of $i$’s in $\{1,2,\ldots,I\}$ of density at least $1-\frac{\varepsilon}{3}$ we have

[TABLE]

for all $(i_{1},\ldots,i_{j})\in S$.

The Van der Corput lemma (Lemma LABEL:vdrCorput of Appendix) finishes the proof.

Note that the set of “good” $i$’s in the interval $\{1,2,\ldots,I\}$ depends on $(i_{1},\ldots,i_{j})\in S$.

Denote

[TABLE]

[TABLE]

Denote $y=a_{1}x+b_{1}m$. Assume that $(a_{1},b_{1})=d$. Denote

[TABLE]

where $a_{p}^{\prime}=\frac{a_{p}}{a_{1}}$, $b_{p}^{\prime}=b_{p}-a_{p}^{\prime}b_{1}$, $2\leq p\leq k$. We rewrite $\tilde{A}$ as follows:

[TABLE]

Here $\phi$ is a bijection of $\mathbb{Z}_{\frac{a_{1}}{d}}$ defined by the identity

[TABLE]

for every $0\leq l\leq\frac{a_{1}}{d}-1$, $Y=a_{1}X$, $a_{p}^{\prime},b_{p}^{\prime}$ as above and $\delta_{X,M}$ accounts for the fact that for small $y$’s and $y$’s close to $Y$ there is a difference between elements that are taken in the expression for $\tilde{A}$ and in the expression on the right hand side of equation (LABEL:tilde_eq). Nevertheless, we have $\delta_{X,M}\rightarrow 0$ if $\frac{M}{X}\rightarrow 0$.

Denote

[TABLE]

It will suffice to prove that there exists $\mathbb{I}(\varepsilon,\delta)\in\mathbb{N}$ such that for every $\mathbb{I}>\mathbb{I}(\varepsilon,\delta)$ we can find $S\subset[-\mathbb{I},\mathbb{I}]^{j}$ of density at least $1-\delta$ with the following property:

There exists $I(\varepsilon,S)\in\mathbb{N}$ such that for every $I>I(\varepsilon,S)$ there exists $M(I)\in\mathbb{N}$ such that for every $M>M(I)$ we can find $X(M)\in\mathbb{N}$ such that for every $X>X(M)$ for a set of $i$’s in $\{1,2,\ldots,I\}$ of density at least $1-\frac{\varepsilon}{3}$ we have

[TABLE]

for all $0\leq l\leq\frac{a_{1}}{d}-1$, for all $(i_{1},\ldots,i_{j})\in S$.

Note that it is enough to prove the latter statement for every particular $l:\,0\leq l\leq\frac{a_{1}}{d}-1$.

Denote the left hand side of inequality (LABEL:sub_eq_lemma) for a fixed $l$ by $\tilde{D}_{l}$.

Introduce new variables $z$ and $n$, such that $y=za_{1}+dl$ and $m=n\frac{a_{1}}{d}+\phi(l)$. We obtain

[TABLE]

[TABLE]

where $sh_{p}=(r_{p},(c_{1}^{p}i_{1},\ldots,c_{j}^{p}i_{j},b_{p}i))$,

$t_{n,z,l}^{p}=\frac{a_{p}(a_{1}z+dl)+(a_{1}b_{p}-a_{p}b_{1})(\frac{a_{1}}{d}n+\phi(l))}{a_{1}}$, $q_{p}=\frac{a_{p}ld+(a_{1}b_{p}-a_{p}b_{1})\phi(l)}{a_{1}}$,

$c_{p}=\frac{a_{1}b_{p}-a_{p}b_{1}}{d}\neq 0$, $Z=\frac{Y}{a_{1}}$ and $N=\frac{Md}{a_{1}}$.

From the conditions on the function $\phi$ it follows that $q_{p}\in\mathbb{Z},\,2\leq p\leq k$.

From the conditions of the lemma we obtain for every $p\neq q,\,\,p,q>1$,

[TABLE]

Therefore, $\tilde{D}_{l}$ can be rewritten as

[TABLE]

where $\phi_{\ell}=\psi^{\ell,j+1}_{r_{\ell}+q_{\ell},(c_{1}^{\ell}i_{1},\ldots,c_{j}^{\ell}i_{j},b_{\ell}i)},\,\,2\leq\ell\leq k$. By the induction hypothesis the following is true.

There exists $\mathbb{I}_{l}(\varepsilon,\delta^{\prime})\in\mathbb{N}$ big enough, such that for every $\mathbb{I}_{l}\geq\mathbb{I}_{l}(\varepsilon,\delta^{\prime})$ there exist a subset $S_{l}\subset[-\mathbb{I}_{l},\mathbb{I}_{l}]^{j+1}$ of density at least $1-\delta^{\prime 2}$ and $N(S_{l},\varepsilon)\in\mathbb{N}$, such that for every $N\geq N(S_{l},\varepsilon)$ there exists $Z(N,S_{l},\varepsilon)\in\mathbb{N}$, such that for every $Z\geq Z(N,S_{l},\varepsilon)$ we have

[TABLE]

for all $(i_{1},\ldots,i_{j},i)\in S_{l}$.

For every $(i_{1},\ldots,i_{j})\in[-\mathbb{I}_{l},\mathbb{I}_{l}]^{j}$ we denote by $S_{i_{1},\ldots,i_{j}}^{l}$ the fiber above $(i_{1},\ldots,i_{j})$:

[TABLE]

Then there exists a set $T_{l}\subset[-\mathbb{I}_{l},\mathbb{I}_{l}]^{j}$ of density at least $1-\delta^{\prime}$, such that for every $(i_{1},\ldots,i_{j})\in T_{l}$ the density of $S_{i_{1},\ldots,i_{j}}^{l}$ is at least $1-\delta^{\prime}$. Let $\varepsilon,\delta>0$. Take $\delta^{\prime}<\min{(\frac{\varepsilon}{6},\delta)}$ and $\mathbb{I}>\max{(I^{\prime}(\varepsilon),\mathbb{I}_{l}(\varepsilon,\delta^{\prime}))}$ ($I^{\prime}(\varepsilon)$ is taken from the van der Corput lemma).

Then it follows by (LABEL:ineq_last) that there exists $M(T_{l},\varepsilon,\delta)\in\mathbb{N}$, such that for every $M\geq M(T_{l},\varepsilon,\delta)$ there exists $X(M,T_{l},\varepsilon,\delta)\in\mathbb{N}$, such that for every $X\geq X(M,T_{l},\varepsilon,\delta)$ the inequality (LABEL:sub_eq_lemma) holds for every fixed $(i_{1},\ldots,i_{j})\in T_{l}$ for a set of $i$’s within the interval $\{1,\ldots,\mathbb{I}\}$ of density at least $1-\frac{\varepsilon}{3}$. The lemma follows from the van der Corput lemma.

$\Box$

Proof of Proposition LABEL:main_prop.

Denote $v_{m}(n)\doteq\xi_{1}(a_{1}n+b_{1}m)\ldots\xi_{k}(a_{k}n+b_{k}m)$. For every $i\in\mathbb{N}$ we introduce $\tilde{A}$ defined by

[TABLE]

Then

[TABLE]

where the functions $\psi^{p,j}$’s are autocorrelation functions of the $\xi_{p}$’s of length $j$.

By Lemma LABEL:autocorrelation2 it follows that for every $\varepsilon>0$ there exists $I(\varepsilon)\in\mathbb{N}$ such that for every $I\geq I(\varepsilon)$ there exist $S\subset\{1,2,\ldots,I\}$ of density at least $1-\frac{\varepsilon}{3}$ and $M(S,\varepsilon)$ such that for every $M\geq M(S,\varepsilon)$ there exists $N(M,S,\varepsilon)$ such that for every $N\geq N(M,S,\varepsilon)$ we have

[TABLE]

The proposition follows from the van der Corput Lemma LABEL:vdrCorput.

$\Box$

3 Probabilistic constructions of WM sets

The goal of this section is to prove the necessity of the conditions of Theorem LABEL:main_thm_lin. The following proposition is the main tool for this task.

Proposition 3.1

Let $a,b\in\mathbb{N}$, $c\in\mathbb{Z}$ such that $a\neq b$. Then there exists a normal set $A$ within which the equation

[TABLE]

is unsolvable, i.e., for every $(x,y)\in A^{2}$ we have $ax\neq by+c$.

Remark 3.1

The proposition is a particular case of Theorem LABEL:main_thm_lin. It is a crucial ingredient in proving the necessity direction of the theorem in general.**

**Proof. ** Let $S\subset\mathbb{N}$. We construct from $S$ a new set $A_{S}$ within which the equation $ax=by+c$ is unsolvable.

Without loss of generality, suppose that $a<b$.

Assume $(a,b)=1$ (the general case follows easily). It follows from $(a,b)=1$ that (LABEL:lin_eq_gen) is solvable. Any solution $(x,y)$ of the equation $ax=by+c$ has restrictions on $x$. Namely, $x\equiv\phi(a,b,c)(\!\!\!\!\mod b)$, where $\phi(a,b,c)\in\{0,1,\ldots,b-1\}$ is determined uniquely. Let us denote $l_{0}\doteq\phi(a,b,c)$. We define inductively a sequence $\{l_{i}\}\subset\mathbb{N}\cup\{0\}$. If a pair $(x,y)$ is a solution of equation (LABEL:lin_eq_gen) and $y\in b^{i}\mathbb{N}+l_{i-1}$ then choose $l_{i}\in\{0,1,\ldots,b^{i+1}-1\}$ such that $x\in b^{i+1}\mathbb{N}+l_{i}$.

Note that from $(a,b)=1$ it follows that $(a,b^{i+1})=1$. It is clear that if $u,v\in\mathbb{N}$ satisfy $(u,v)=1$ then for any $w\in\mathbb{Z}$ there exists a solution $(x,y)\in\mathbb{N}^{2}$ of the equation $ux=vy+w$. The latter implies that there exist $x\in\mathbb{N},y\in b^{i}\mathbb{N}+l_{i-1}$ such that $ax=by+c$. Any such $x$ should be a member of $b^{i+1}\mathbb{N}+l_{i}$. Note that $l_{i}$ and $l_{i-1}$ are connected by the identity

[TABLE]

In addition, if $x\in\mathbb{N}$ is given then the equation

[TABLE]

has at most one solution $y\in\{0,1,\ldots,b^{i}-1\}$.

We define sets $H_{i}\doteq b^{i}\mathbb{N}+l_{i-1}\,;\,i\in\mathbb{N}$. We prove that for every $i\in\mathbb{N},\,H_{i+1}\subset H_{i}$. All elements of $H_{i+1}$ are in the same class modulo $b^{i+1}$, therefore all elements of $H_{i+1}$ are in the same class modulo $b^{i}$. So, if we show for some $x\in H_{i+1}$ that $x\equiv l_{i-1}(\!\!\!\!\mod b^{i})$ then we are done. For $i=1$ we know that if $y\in\mathbb{N}$ then any $x\in\mathbb{N}$ such that $(x,y)$ is a solution of the equation (LABEL:lin_eq_gen) has to be in $H_{1}$. Take $x\in H_{2}$ such that there exists $y\in H_{1}$ with $ax=by+c$. Then $x\in H_{1}$. Therefore, we have shown that $H_{2}\subset H_{1}$. For $i>1$ there exists $x\in H_{i+1}$ such that there exists $y\in H_{i}$ with $ax=by+c$. By induction $H_{i}\subset H_{i-1}$. Therefore, the latter $y$ is in $H_{i-1}$. Therefore, by construction of $l_{i}$’s we have that $x\in H_{i}$. This shows $H_{i+1}\subset H_{i}$. We define sets $B_{i};\,0\leq i<\infty$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Clearly we have $B_{i}\cap B_{j}=\emptyset\,\,,\forall i\neq j$ and $|\mathbb{N}\setminus(\cup_{i=0}^{\infty}B_{i})|=|\cap_{i=1}^{\infty}H_{i}|\leq 1$. The latter is because for every $i$ the second element (in the increasing order) of $H_{i}$ is $\geq b^{i}$.

We define $A_{S}=\bigcup_{i=0}^{\infty}A_{i}$, where $A_{i}$’s are defined in the following manner:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here it is worthwhile to remark that for every $i,\,\,B_{i}=A_{i}\cup C_{i}$. Therefore $A_{S}\subset\cup_{i=0}^{\infty}B_{i}$.

If for some $i\geq 1$ we have $y\in A_{i}\subset B_{i}=H_{i}\setminus H_{i+1}$, then any $x$ with $ax=by+c$ satisfies

[TABLE]

From $(a,b^{i+1})=1$ it follows that there exists a unique solution $x$ modulo $b^{i+1}$. By identity (LABEL:equiv_ls) we have

[TABLE]

Thus $x\in H_{i+1}$.

If $x\in H_{i+2}$, then

[TABLE]

Thus we have

[TABLE]

By uniqueness of a solution ( $y$ ) modulo $b^{i+1}$ we get

[TABLE]

Thus $y\in H_{i+1}$. We have a contradiction, which shows that $x\in H_{i+1}\setminus H_{i+2}=B_{i+1}$.

The same argument works for $y\in A_{0}\subset B_{0}$ and it shows that any $x$ with $ax=by+c$ satisfies $x\in B_{1}$.

So, if $y\in A_{i}$ ($i\geq 0$) then any $x$ with $ax=by+c$ should satisfy $x\in B_{i+1}$. By construction of $A_{S}$, $x\not\in A_{S}$. Thus equation (LABEL:lin_eq_gen) is not solvable in $A_{S}$.

We make the following claim:

For almost every subset $S$ of $\mathbb{N}$ the set $A_{S}$ is a normal set.

(The probability measure on subsets of $\mathbb{N}$ considered here is the product on $\{0,1\}^{\infty}$ of probability measures $(\frac{1}{2},\frac{1}{2})$.)

The tool for proving the claim is the following easy lemma (for a proof see Appendix, Lemma LABEL:norm_lemma).

A subset $A$ of natural numbers is a normal set if and only if for any $k\in(\mathbb{N}\cup\{0\})$ and any $i_{1}<i_{2}<\ldots<i_{k}$ we have

[TABLE]

where $\chi_{A}(n)\doteq 2\cdot 1_{A}(n)-1$.

First of all, we denote $T_{N}=\frac{1}{N}\sum_{n=1}^{N}\chi_{A_{S}}(n)\chi_{A_{S}}(n+i_{1})\ldots\chi_{A_{S}}(n+i_{k})$. Because of randomness of $S$, $T_{N}$ is a random variable. We will prove that $\sum_{N=1}^{\infty}E(T_{N^{2}}^{2})<\infty$ and this will imply by Lemma LABEL:tec_lemma that $T_{N}\rightarrow_{N\rightarrow\infty}0$ for almost every $S\subset\mathbb{N}$.

[TABLE]

Adding (removing) of a finite set to (from) a normal set does not affect the normality of the set. The set $\cup_{i}B_{i}$ might differ from $\mathbb{N}$ by at most one element ($|\cap_{i=1}^{\infty}H_{i}|\leq 1$). This possible element does not affect the normality of $A_{S}$ and we assume without loss of generality that $\cap_{i=1}^{\infty}H_{i}=\emptyset$, thus $\mathbb{N}=\cup_{i=0}^{\infty}B_{i}$. For every number $n\in\mathbb{N}$ we define the chain of $n$, $Ch(n)$, to be the following finite sequence:

If $n\in B_{0}$, then $Ch(n)=(n)$.

If $n\in B_{1}$, then two situations are possible. In the first one there exists a unique $y\in B_{0}$ such that $an=by+c$. We set $Ch(n)=(n,y)=(n,Ch(y))$. In the second situation we can not find such $y$ from $B_{0}$ and we set $Ch(n)=(n)$.

If $n\in B_{i+1}$, then again two situations are possible. In the first one there exists $y\in B_{i}$ such that $an=by+c$. In this case we set $Ch(n)=(n,Ch(y))$. In the second situation there is no such $y$ from $B_{i}$. In this case we set $Ch(n)=(n)$. We define $l(n)$ to be the length of $Ch(n)$.

For every $n\in\mathbb{N}$ we define the ancestor of $n$, $a(n)$, to be the last element of the chain of $n$ (of $Ch(n)$). To determine whether or not $n\in A_{S}$ will depend on whether $a(n)\in S$. The exact relationship depends on the $i$ for which $n\in B_{i}$ and on the $j$ for which $a(n)\in B_{j}$ or in other words on the length of $Ch(n)$: $\chi_{A_{S}}(n)=(-1)^{i-j}\chi_{S}(a(n))=(-1)^{l(n)-1}\chi_{S}(a(n))$.

We say that $n$ is a descendant of $a(n)$.

It is clear that $E(\chi_{A_{S}}(n_{1})\ldots\chi_{A_{S}}(n_{k}))\neq 0$ ($E(\chi_{A_{S}}(n_{1})\ldots\chi_{A_{S}}(n_{k}))\in\{0,1\}$) if and only if every number $a(n_{i})$ occurs an even number of times among numbers $a(n_{1}),a(n_{2}),\ldots,a(n_{k})$.

We bound the number of $n,m$’s inside the square $[1,N]\times[1,N]$ such that $E(\chi_{A_{S}}(n)\chi_{A_{S}}(n+i_{1})\ldots\chi_{A_{S}}(n+i_{k})\chi_{A_{S}}(m)\chi_{A_{S}}(m+i_{1})\ldots\chi_{A_{S}}(m+i_{k}))\neq 0$.

For a given $n\in[1,N]$ we count all $m$’s inside $[1,N]$ such that for the ancestor of $n$ there will be a chance to have a twin among the ancestors of $n+i_{1},\ldots,n+i_{k},m,m+i_{1},\ldots,m+i_{k}$.

First of all it is obvious that in the interval $[1,N]$ for a given ancestor there can be at most $\log_{\frac{b}{a}}N+C_{1}$ descendants, where $C_{1}$ is a constant. For all but a constant number of $n$’s it is impossible that among $n+i_{1},\ldots,n+i_{k}$ there is the same ancestor as for $n$. Therefore we should focus on ancestors of the set $\{m,m+i_{1},\ldots,m+i_{k}\}$. For a given $n$ we might have at most $(k+1)(\log_{\frac{b}{a}}N+C_{1})$ options for the number $m$ to provide that one of the elements of $\{m,m+i_{1},\ldots,m+i_{k}\}$ has the same ancestor as $n$. Therefore for most of $n\in[1,N]$ (except maybe a bounded number $C_{2}$ of $n$’s which depends only on $\{i_{1},\ldots,i_{k}\}$ and doesn’t depend on $N$) we have at most $(k+1)(\log_{\frac{b}{a}}N+C_{1})$ possibilities for $m$’s such that

[TABLE]

Thus we have

[TABLE]

where $C_{3}$ is a constant. This implies

[TABLE]

Therefore $T_{N^{2}}\rightarrow_{N\rightarrow\infty}0$ for almost every $S\subset\mathbb{N}$. By Lemma LABEL:tec_lemma it follows that $T_{N}\rightarrow_{N\rightarrow\infty}0$ almost surely.

In the general case, where $a,b$ are not relatively prime, if $c$ satisfies (LABEL:lin_eq_gen) then it should be divisible by $(a,b)$. Therefore by dividing the equation (LABEL:lin_eq_gen) by $(a,b)$ we reduce the problem to the previous case.

$\Box$

We use the following notation:

Let W be a subset of $\mathbb{Q}^{n}$. Then for any increasing subsequence $I=(i_{1},\ldots,i_{p})\subset\{1,2,\ldots,n\}$ we define

[TABLE]

We recall the notion of a cone.

Definition 3.1

*A subset $W\subset\mathbb{Q}^{n}$ is called a cone if

(a) $\forall w_{1},w_{2}\in W$ we have $w_{1}+w_{2}\in W$

(b) $\forall\alpha\in\mathbb{Q}:\,\alpha\geq 0$ and $\forall w\in W$ we have $\alpha w\in W$.*

The next step involves an algebraic statement with a topological proof which we have to establish.

Lemma 3.1

Let $W$ be a non-trivial cone in $\mathbb{Q}^{n}$ which has the property that for every two vectors $\vec{a}=\{a_{1},a_{2},\ldots,a_{n}\}^{t},\vec{b}=\{b_{1},b_{2},\ldots,b_{n}\}^{t}\in W$ there exist two coordinates $1\leq i<j\leq n$ (depend on the choice of $\vec{a},\vec{b}$) such that

[TABLE]

There exist two coordinates $i<j$ such that the projection of $W$ on these two coordinates is of dimension $\leq 1$ ($dim_{\mathbb{Q}}\,Span\,Proj_{(i,j)}W\leq 1$).

**Proof. ** First of all $W$ has positive volume in $V=SpanW$ (Volume is Haar measure which normalized by assigning measure one to a unit cube and $W$ contains a parallelepiped). Fix an arbitrary non-zero element $\vec{x}\in W$. For every $i,j:\,1\leq i<j\leq n$ we define the subspace

[TABLE]

From the assumptions of the lemma it follows that

[TABLE]

For every $i\neq j$ we obviously have that the volume of $U_{i,j}$ is either zero or $U_{i,j}=V$. If we assume that the statement of the lemma does not hold then $U_{i,j}\neq V,\,\forall i\neq j$, and thus the volume of $U_{i,j},\,\forall i\neq j$ is zero. We get a contradiction because a finite union of sets with zero volume cannot be equal to a set with positive volume.

$\Box$

**Proof. **(of Theorem LABEL:main_thm_lin, $\Rrightarrow$)

Assume that an affine subspace $\mathbb{A}$ of $\mathbb{Q}^{k}$ intersects $A^{k}$ for any WM set $A\subset\mathbb{N}$.

First of all, we shift the affine space to obtain a vector subspace, denote it by $U$. The linear space $U$ must contain vectors with all positive coordinates, since $\mathbb{A}\cap A^{k}$ must be infinite.

Denote by $W=\{\vec{v}\in U\,|\,\langle\vec{v},\vec{e_{i}}\rangle\geq 0\,,\,\forall\,i:\,1\leq i\leq k\}$. $W$ is a non-trivial cone.

Assume that for every $\vec{a}=(a_{1},\ldots,a_{k})^{t},\vec{b}=(b_{1},\ldots,b_{k})^{t}\in W$ we have that $\exists i,j:\,\,1\leq i<j\leq k$ such that

[TABLE]

Then by Lemma LABEL:3_algebraic_lemma we deduce that there exist maximal subsets of coordinates $F_{1},\ldots,F_{l}$ (one of them, assume $F_{1}$, should have at least two coordinates) such that for every $r\in\{1,2\ldots,l\}$ we have $V_{F_{r}}\doteq SpanW_{F_{r}}$ is one dimensional.

We fix $r\,:\,1\leq r\leq l$. We show that the projection on $F_{r}$ of $W+\vec{f}$ is on a diagonal, where $\vec{f}\in\mathbb{Z}^{k}$ is such that $U+\vec{f}=\mathbb{A}$. If the projection of $W$ on $F_{r}$ is not on a diagonal then there exist two coordinates $i<j$ from $F_{r}$ such that $W_{(i,j)}=\{(ax,bx)\,|\,x\in\mathbb{N}\}$ for some $a\neq b$ natural numbers. Therefore the projection of $\mathbb{A}$ on $(i,j)$ has the form $\{(ax+f_{1},bx+f_{2})\,|\,x\in\mathbb{N}\}$, where $f_{1},f_{2}$ are integers. From Proposition LABEL:3_constr_prop it follows that for any $a,b,c$, where $a\neq b$, there exists a WM set $A$ (even a normal set) such that the equation $ax=by+c$ is not solvable within $A$. This proves the existence of a WM set $A_{0}$ such that for every $x\in\mathbb{Z}$ we have $(ax+f_{1},bx+f_{2})\not\in A_{0}^{2}$ (introduce the new variables $z_{1},z_{2}$ by $(z_{1},z_{2})=(ax_{1}+f_{1},bx+f_{2})$ and take a normal set $A_{0}$ such that the equation $az_{2}=bz_{1}+(af_{2}-bf_{1})$ is unsolvable within $A_{0}$).

Thus $\forall i,j\in F_{r}:\,\,W_{(i,j)}=\{(ax,ax)\,|\,x\in\mathbb{N}\}$.

To prove that a shift is the same for all coordinates in $F_{r}$ we merely should know that for any natural number $c$ there exists a WM set $A_{c}$ such that inside $A_{c}$ the equation $x-y=c$ is not solvable. The last statement is easy to verify.

Let $j_{r}\in F_{r}$, $\forall 1\leq r\leq l$. Denote $I=(j_{1},\ldots,j_{l})$. We have proved that there exist $g_{1},\ldots,g_{l}\in\mathbb{N}$, $c_{1},\ldots,c_{l}\in\mathbb{Z}$ such that

[TABLE]

It is clear that we can find $\vec{a},\vec{b}$ which satisfy all the requirements of Theorem LABEL:main_thm_lin. This completes the proof.

$\Box$

Remark 3.2

*We have proved that if an affine subspace $\mathbb{A}\subset\mathbb{Q}^{k}$ intersects $A^{k}$ for any normal set $A\subset\mathbb{N}$, then there exist $\vec{a},\vec{b}\in\mathbb{N}^{k}$ and a partition $F_{1},\ldots,F_{l}$ of $\{1,2,\ldots,k\}$ such that:

(a) $\forall r:\,1\leq r\leq l$ and $\forall i\in F_{r},\forall j\not\in F_{r}$ we have*

[TABLE]

(b)* $\exists\vec{f}\in\mathbb{Z}^{k}$ such that the set $\{n\vec{a}+m\vec{b}+\vec{f}\,|\,n,m\in\mathbb{N}\}$ is in $\mathbb{A}$.*

Thus, we have proved the direction “$\Rrightarrow$” of Theorem LABEL:normal_theorem.

4 Comparison with Rado’s Theorem

We recall that the problem of solvability of a system of linear equations in one cell of any finite partition of $\mathbb{N}$ was solved by Rado in [10]. Such systems of linear equations are called partition-regular. We show that partition-regular systems are solvable within every WM set by use of Theorem LABEL:main_thm_lin. It is important to note that solvability of partition-regular linear systems of equations within WM sets can be shown directly (without use of Theorem LABEL:main_thm_lin) by use of the technique of Furstenberg and Weiss that was developed in their dynamical proof of Rado’s theorem (see [8]).

First of all we describe Rado’s regular systems.

Definition 4.1

A rational $p\times q$ matrix $(a_{ij})$ is said to be of level $l$ if the index set $\{1,2,\ldots,q\}$ can be divided into $l$ disjoint subsets $I_{1},I_{2},\ldots,I_{l}$ and rational numbers $c_{j}^{r}$ may be found for $1\leq r\leq l$ and $1\leq j\leq q$ such that the following relationships are satisfied:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for $i=1,2,\ldots,p$.

Theorem 4.1

(Rado) A system of linear equations is partition-regular if and only if for some $l$ the matrix $(a_{ij})$ is of level $l$ and it is homogeneous, i.e. a system of the form

[TABLE]

The following claim is the main result of this section.

Proposition 4.1

A partition-regular system is solvable in every WM set.

**Proof. ** Let a system $\sum_{j=1}^{q}a_{ij}x_{j}=0,i=1,2,\ldots,p$ be partition-regular. We will use the fact that the system is solvable for any finite partition of $\mathbb{N}$. First of all, the set of solutions of a partition-regular system is a subspace of $\mathbb{Q}^{q}$; denote it by $V$. It is obvious that $V$ contains vectors with all positive components. If for some $1\leq i<j\leq q$ we have $Proj_{i,j}^{+}V$ (where $Proj_{i,j}^{+}V=\{(x,y)|x,y\geq 0\hskip 7.22743pt\&\hskip 7.22743pt\exists\vec{v}\in V:\,<\vec{v},\vec{e_{i}}>=x\,,\,<\vec{v},\vec{e_{j}}>=y\}$) is contained in a line, then $Proj_{i,j}^{+}V$ is diagonal, i.e. it is contained in $\{(x,x)|x\in\mathbb{Q}\}$. Otherwise, we can generate a partition of $\mathbb{N}$ into two disjoint sets $S_{1},S_{2}$ such that no $S_{1}^{q}$ and no $S_{2}^{q}$ intersects $V$:

This partition is constructed by an iterative process. Without loss of generality we may assume that the line is $x=ny$, where $n\in\mathbb{N}$. The general case is treated in the simillar way. We start with $S_{1}=S_{2}=\emptyset$. Let $1\in S_{1}$.

We “color” the infinite geometric progression $\{n^{m}\,|\,m\in\mathbb{N}\}$ (adding elements to either $S_{1}$ or $S_{2}$) in such way that there is no $(x,y)$ on the line from $S_{1}^{2},S_{2}^{2}$. Then we take a minimal element from $\mathbb{N}$ which is still uncolored. Call it $a$. Add $a$ to $S_{1}$. Next, “color” $\{an^{m}\,|\,m\in\mathbb{N}\}$.

Continuing in this fashion, we obtain the desired partition of $\mathbb{N}$.

This contradicts the assumption that the given system is partition-regular.

Let $F_{1},\ldots,F_{l}$ be a partition of $\{1,2,\ldots,k\}$ such that for every $r\in\{1,\ldots,l\}$ we have for every $i\neq j\,,\,i,j\in F_{r}:\,\hbox{\rm dim}\,_{\mathbb{Q}}Span(Proj_{i,j}^{+}V)=1$, and for every $r:\,1\leq r\leq l$, every $i\in F_{r}$ .and for every $j\not\in F_{r}$ we have $\hbox{\rm dim}\,_{\mathbb{Q}}Span(Proj_{i,j}^{+}V)=2$. For every $r:1\leq r\leq l$ we choose arbitrarily one representative index within $F_{r}$ and denote it by $j_{r}$ ($j_{r}\in F_{r}$).

Then there exist $g_{1},\ldots,g_{l}\in\mathbb{N}$ such that

[TABLE]

The latter ensures that there exist vectors $\vec{a},\vec{b}\in V$ which satisfy all the requirements of Theorem LABEL:main_thm_lin and, therefore, the system is solvable in every WM set.

$\Box$

5 Appendix

In this section we prove all technical lemmas and propositions that were used in the paper.

We start with the key lemma which is a finite modification of Bergelson’s lemma in [1]. Its origin is in a lemma of van der Corput.

Lemma 5.1

*Suppose $\varepsilon>0$ and $\{u_{j}\}_{j=1}^{\infty}$ is a family of vectors in Hilbert space, such that $\|u_{j}\|\leq 1\,\rm(1\leq j\leq\infty\rm)$. Then there exists $I^{\prime}(\varepsilon)\in\mathbb{N}$, such that for every $I\geq I^{\prime}(\varepsilon)$ there exists $J^{\prime}(I,\varepsilon)\in\mathbb{N}$, such that the following holds:

For $J\geq J^{\prime}(I,\varepsilon)$ for which we obtain*

[TABLE]

for a set of $i$’s in the interval $\{1,\ldots,I\}$ of density $1-\frac{\varepsilon}{3}$ we have

[TABLE]

**Proof. ** For an arbitrary $J$ define $u_{k}=0$ for every $\textrm{k}<1$ or $k>J$. The following is an elementary identity:

[TABLE]

Therefore, the inequality $\left\|\sum_{i=1}^{N}u_{i}\right\|^{2}\leq N\sum_{i=1}^{N}\left\|u_{i}\right\|^{2}$ yields

[TABLE]

[TABLE]

where $\Sigma_{1}=I\sum^{J}_{j=1}\left\|u_{j}\right\|^{2}$ by the aforementioned elementary identity and $\Sigma_{2}=\sum^{I-1}_{h=1}(I-h)\sum^{J}_{j=1}\langle u_{j},u_{j+h}\rangle$. The last expression is obtained by rewriting $\Sigma_{2}$, where $h=r-s$. By dividing the foregoing inequality by $I^{2}J^{2}$ we obtain

[TABLE]

Choose $I^{\prime}(\varepsilon)\in\mathbb{N}$, such that $\frac{12}{\varepsilon}\leq I^{\prime}(\varepsilon)\leq\frac{12}{\varepsilon}+1$. Then for every $I\geq I^{\prime}(\varepsilon)$ we have $\frac{1}{I}+\frac{5\varepsilon}{6}\leq\frac{11\varepsilon}{12}$. There exists $J^{\prime}(I,\varepsilon)\in\mathbb{N}$, such that for every $J\geq J^{\prime}(I,\varepsilon)$: $\frac{J+I}{J}<\frac{12}{11}$. As a result, for every $I\geq I^{\prime}(\varepsilon)$ there exists $J^{\prime}(I,\varepsilon)$, such that for every $J\geq J^{\prime}(I,\varepsilon)$

[TABLE]

$\Box$

The next proposition was used in Section LABEL:lin_proof_suff.

Proposition 5.1

Let $A\subset\mathbb{N}$ be a WM-set. Then for every integer $a>0$ and every integers $b_{1},b_{2},\ldots,b_{k}$

[TABLE]

[TABLE]

where $\xi\doteq 1_{A}-\mathrm{d}(A)$.

**Proof. ** Consider the weak-mixing measure preserving system $(X_{\xi},\mathbb{B},\mu,T)$.

The left side of the equation in the proposition is $\int_{X_{\xi}}T^{b_{1}}fT^{b_{2}}f\ldots T^{b_{k}}fd\mu$, where $f(\omega)\doteq\omega_{0}$ for every infinite sequence inside $X_{\xi}$. We make use of the notion of disjointness of measure preserving systems. By [6] we know that every weak-mixing system is disjoint from any Kronecker system which is a compact monothethic group with Borel $\sigma$-algebra, the Haar probability measure, and the shift by a chosen element of the group. In particular, every weak-mixing system is disjoint from the measure preserving system $(\mathbb{Z}_{a},\mathbb{B}_{\mathbb{Z}_{a}},S,\nu)$, where $\mathbb{Z}_{a}=\mathbb{Z}/a\mathbb{Z}$, $S(n)\doteq n+1(\mod a)$. The measure and the $\sigma$-algebra of the last system are uniquely determined. Therefore, from Furstenberg’s theorem (see [6], Theorem I.6) it follows that the point $(\xi,0)\in X_{\xi}\times\mathbb{Z}_{a}$ is a generic point of the product system $(X_{\xi}\times\mathbb{Z}_{a},\mathbb{B}\times\mathbb{B}_{\mathbb{Z}_{a}},T\times S,\mu\times\nu)$. Thus, for every continuous function $g$ on $X_{\xi}\times\mathbb{Z}_{a}$ we obtain

[TABLE]

Let $g(x,m)\doteq f(x)1_{0}(m)$, which is obviously continuous on $X_{\xi}\times\mathbb{Z}_{a}$. Then genericity of the point $(\xi,0)$ yields

[TABLE]

[TABLE]

Taking instead of the function $f$ the continuous function $T^{b_{1}}fT^{b_{2}}f\ldots T^{b_{k}}f$ in the definition of $g$ finishes the proof.

$\Box$

The next two lemmas are very useful for constructing normal sets with specifical properties.

Lemma 5.2

Let $A\subset\mathbb{N}$. Let $\lambda(n)=21_{A}(n)-1$. Then $A$ is a normal set $\Leftrightarrow$ for any $k\in(\mathbb{N}\cup\{0\})$ and any $i_{1}<i_{2}<\ldots<i_{k}$ we have

[TABLE]

**Proof. ** “$\Rightarrow$” If $A$ is normal then any finite word $w\in\{-1,1\}^{*}$ has the “right” frequency $\frac{1}{2^{|w|}}$ inside $w_{A}$. This guarantees that “half of the time” the function $\lambda(n)\lambda(n+i_{1})\ldots\lambda(n+i_{k})$ equals $1$ and “half of the time” is equal to $-1$. Therefore we get the desired conclusion.

“$\Leftarrow$” Let $w$ be an arbitrary finite word of plus and minus ones: $w=a_{1}a_{2}\ldots a_{k}$ and we have to prove that $w$ occurs in $w_{A}$ with the frequency $2^{-k}$. For every $n\in\mathbb{N}$ the word $w$ occurs in $1_{A}$ and starting from $n$ if and only if

[TABLE]

The latter is equivalent to the following

[TABLE]

The frequency of $w$ within $1_{A}$ is equal to

[TABLE]

The limit is equal to $\frac{1}{2^{k}}$.

$\Box$

Lemma 5.3

Let $\{a_{n}\}$ be a bounded sequence. Let $T_{N}=\frac{1}{N}\sum_{n=1}^{N}a_{n}$. Then $T_{N}$ converges to a limit $t$ $\Leftrightarrow$ there exists a sequence of increasing indices $\{N_{i}\}$ such that $\frac{N_{i}}{N_{i+1}}\rightarrow 1$ and $T_{N_{i}}\rightarrow_{i\rightarrow\infty}t$.