On Factoring Groups into Thin Subsets

Abstract

A subset X of a group G is called thin if, for every finite subset F of G, there exists a finite subset H of G such that $Fx\cap Fy=\varnothing$, $xF\cap yF=\varnothing$ for all distinct $x,y\in X\backslash H$. We prove that every countable topologizable group G can be factorized $G=AB$ into thin subsets $A,B$.

1. Introduction

Let G be a group, and ${\left[G\right]}^{<\omega }$ denote the set of all finite subsets of G. A subset X of is called:

• left thin if, for every $F\in {\left[G\right]}^{<\omega }$, there exists $H\in {\left[G\right]}^{<\omega }$ such that $Fx\cap Fy=\varnothing$ for all distinct $x,y\in X\backslash H$;
• right thin if, for every $F\in {\left[G\right]}^{<\omega }$, there exists $H\in {\left[G\right]}^{<\omega }$ such that $xF\cap yF=\varnothing$ for all distinct $x,y\in X\backslash H$;
• thin if X is left and right thin.

The notion of left thin subsets was introduced in [1]. For motivation to study left thin, right thin and thin subsets and some results and references, see Comments and surveys [2,3,4,5]. In asymptology, thin subsets play the part of discrete subsets (see Comments 1 and 2).

We recall that the product $AB$ of subsets $A,B$ of a group G is a factorization if $G=AB$ and each element $g\in G$ has the unique representation $g=ab$, $a\in A$, $b\in B$ (equivalently, the subsets $\{aB:a\in A\}$ are pairwise disjoint). For factorizations of groups into subsets, see [6].

Our goal is to prove the following theorem. By a countable set, we mean a countably infinite set. The group topology $\tau$ is supposed to be Hausdorff.

Theorem 1.

Let $(G,\tau )$ be a non-discrete countable topological group. Then G can be factorized $G=AB$ into thin subsets $A,B$.

2. Proof

Proof of Theorem 1.

Let $G=\{g_n:n<\omega \}$, $g_0=e$, e is the identity of G, $F_n=\{g_i:i\le n\}$.

Given two sequences ${\left(a_n\right)}_{n<\omega }$, ${\left(b_n\right)}_{n<\omega }$ in G, we denote

$$A_n=\{a_i,a_i^{-1}:i\le n\},B_n=\{b_i:i\le n\},A={\cup }_{n<\omega }{A}_n,B={\cup }_{n<\omega }{B}_n.$$

We want to choose ${\left(a_n\right)}_{n<\omega }$, ${\left(b_n\right)}_{n<\omega }$ so that $AB$ is a factorization of G and $A,B$ are thin.

Let $X,Y$ be subsets of G. We say that $XY$ is a partial factorization of G if the subsets $\{Xy:y\in Y\}$ are pairwise disjoint (equivalently, the subsets $\{Yx:x\in X\}$ are pairwise disjoint).

We put $a_0=e$, $b_0=e$ and suppose that $a_0,\cdots ,a_n$ and $b_0,\cdots ,b_n$ have been chosen so that the following conditions are satisfied

$\left(1\right)$$A_n{B}_n$ is a partial factorization of G and $g_n\in A_n{B}_n$;

$\left(2\right)$$F_i{b}_i\cap F_j{b}_j=\varnothing$, $b_i{F}_i\cap b_j{F}_j=\varnothing$ for all distinct $i,j\in \{0,\cdots ,n\}$;

$\left(3\right)$$F_i{a}_i\cap F_j{a}_j=\varnothing$, $a_i{F}_i\cap a_j{F}_j=\varnothing$, $F_i{a}_i^{-1}\cap F_j{a}_j^{-1}=\varnothing$, $a_i^{-1}{F}_i\cap a_j^{-1}{F}_j=\varnothing$ and

$F_i{a}_i^{-1}\cap F_j{a}_j,$$a_i^{-1}{F}_i\cap a_j{F}_j=\varnothing$ for all distinct $i,j\in \{0,\cdots ,n\}$;

$\left(4\right)$ if $a_i\ne a_i^{-1}$ then $F_i{a}_i\cap F_i{a}_i^{-1}=\varnothing$, $a_i{F}_i\cap a_i^{-1}{F}_i=\varnothing$, $i\in \{0,\cdots ,n\}$.

We take the first element $g_m\in G\backslash A_n{B}_n$, put $g=g_m$ and show that there exists a symmetric neighborhood U of e such that

$\left(5\right)$$(A_n\cup \{x,x^{-1}\})(B_n\cup \left\{xg\right\})$ is a partial factorization for each $x\in U\backslash \left\{e\right\}$.

We choose a symmetric neighborhood V of e such that $(A_n\cup \{x,x^{-1}\})B_n$ is a partial factorization of G for each $x\in V\backslash \left\{e\right\}$.

Then we use $A_n=A_n^{-1}$, $g\in G\backslash A_n{B}_n$ and $e\in A_n\cap B_n$ to choose a symmetric neighborhood U of e such that $U\subset V$ and

$$(A_n\cup \{x,x^{-1}\})B_n\cap (A_n\cup \{x,x^{-1}\})xg=\varnothing ,$$

equivalently, $A_n{B}_n\cap A_{n}xg=\varnothing$, $A_n{B}_n\cap \{x,x^{-1}\}xg=\varnothing$, $\{x,x^{-1}\}{B}_n\cap A_{n}xg=\varnothing$, $\{x,x^{-1}\}{B}_n\cap \{x,x^{-1}\}xg=\varnothing$ for each $x\in U\backslash \left\{e\right\}$, so we get $\left(5\right)$. By the continuity of the group operations, the latter is possible because these 4 equalities hold for $x=e$.

If the set $\{x\in U:x^2=e\}$ is infinite then we use $\left(5\right)$ and choose $a_{n+1}\in U$, $a_{n+1}=a_{n+1}^{-1}$ and $b_{n+1}=a_{n+1}g$ to satisfy $\left(1\right)$–$\left(3\right)$ with $n+1$ in place of n. Otherwise, we choose $a_{n+1}\in U$, $a_{n+1}\ne a_{n+1}^{-1}$ and $b_{n+1}=a_{n+1}g$ to satisfy $\left(1\right)$–$\left(4\right)$.

After $\omega$ steps, we get the desired factorization $G=AB$. □

3. Comments

1. Given a set X, a family $\mathcal{E}$ of subsets of $X\times X$ is called a coarse structure on X if

• each $E\in \mathcal{E}$ contains the diagonal ${△}_X:=\{(x,x):x\in X\}$ of X;
• if E, $E^{\prime }\in \mathcal{E}$ then $E\circ E^{\prime }\in \mathcal{E}$ and $E^{-1}\in \mathcal{E}$, where $E\circ E^{\prime }=\{(x,y):\exists z((x,z)\in E,(z,y)\in E^{\prime })\}$, $E^{-1}=\{(y,x):(x,y)\in E\}$;
• if $E\in \mathcal{E}$ and ${△}_X\subseteq E^{\prime }\subseteq E$ then $E^{\prime }\in \mathcal{E}$.

Elements $E\in \mathcal{E}$ of the coarse structure are called entourages on X.

For $x\in X$ and $E\in \mathcal{E}$ the set $E\left[x\right]:=\{y\in X:(x,y)\in \mathcal{E}\}$ is called the ball of radius E centered at x. Since $E={\bigcup }_{x\in X}(\left\{x\right\}\times E\left[x\right])$, the entourage E is uniquely determined by the family of balls $\left\{E\right[x]:x\in X\}$. A subfamily ${\mathcal{E}}^{\prime }\subseteq \mathcal{E}$ is called a base of the coarse structure $\mathcal{E}$ if each set $E\in \mathcal{E}$ is contained in some $E^{\prime }\in {\mathcal{E}}^{\prime }$.

The pair $(X,\mathcal{E})$ is called a coarse space [7] or a ballean [8,9].

A subset B of X is called bounded if $B\subseteq E\left[x\right]$ for some $E\in \mathcal{E}$ and $x\in X$. A subset Y of X is called discrete if, for every $E\in \mathcal{E}$, there exists a bounded subset B such that $E\left[x\right]\cap E\left[y\right]=\varnothing$ for all distinct $x,y\in Y\backslash B$.

2. Formally, coarse spaces can be considered as asymptotic counterparts of uniform topological spaces. However, actually, this notion is rooted in geometry, geometrical group theory and combinatorics (see [7,8,10,11]).

Given a group G, we denote by ${\mathcal{E}}_l$ and ${\mathcal{E}}_r$ the coarse structures on G with the bases

$$\{\{(x,y):x\in Fy\}:F\in {\left[G\right]}^{<\omega },e\in F\},\{\{(x,y):x\in yF\}:F\in {\left[G\right]}^{<\omega },e\in F\}$$

and note that a subset A of G is left (resp. right) thin if and only if A is discrete in the coarse space $(G,{\mathcal{E}}_l)$ (resp. $(G,{\mathcal{E}}_r)$ ).

3. By [12], every countable group G has a thin subset A such that $G=A{A}^{-1}$. By [13], every countable topological group G has a closed discrete subset A such that $G=A{A}^{-1}$. For thin subsets of topological groups and factorizations into dense subsets, see [14,15].

4. Can every countable group G be factorized $G=AB$ into infinite subsets $A,B$? By Theorem 1, an answer to the following question could be negative only in the case of a non-topologizable group G.

On the other hand, analyzing the proof, one can see that Theorem 1 remains true if all mappings $x⟼xg$, $x⟼gx$, $g\in G$, $x⟼x^{-1}$ and $x⟼x^2$ are continuous at e. By [16], every countable group G admits a non-discrete Hausdorff topology in which all shifts and the inversion $x⟼x^{-1}$ are continuous.