## Sets of recurrence of $$\mathbb Z^m$$-actions and properties of sets of differences in $$\mathbb Z^m$$.(English)Zbl 0579.10029

Let us say that a set $$B\subset \mathbb Z^m$$ is a block if it is a product of intervals. The width of a block is the length of its shortest edge. A set $$A\subset \mathbb Z^m$$ is said to have positive upper Banach density if for some sequence of blocks $$B_n$$ whose widths approach infinity, $$| A\cap B_n| /| B_n| >\delta >0.$$ The set of differences $$A-A$$ of a set $$A\subset \mathbb Z^m$$ is defined by $$A-A=\{a_ 1-a_ 2| \quad a_ 1,a_ 2\in A\}.$$ The author applies an ergodic-theoretical technique to obtain some new properties of sets of differences. In particular, the following two theorems are proved:
Corollary 3.1.1. Let $$A\subset \mathbb Z^2$$ and suppose that $$A$$ has positive upper Banach density. Then there exists $$B\subset\mathbb Z$$ such that $$B$$ has positive upper density and $$A-A\supset B\times B,$$ where $$B\times B$$ is the Cartesian square of $$B$$.
Corollary 3.1.2. If $$A\subset\mathbb Z$$ has positive upper Banach density then there exists a set $$B\subset\mathbb Z$$ of positive upper density such that $$A-A\supset B+B=\{b_ 1+b_ 2| b_ 1,b_ 2\in B\}.$$
Remark. The following two questions are open:
Question 1. Let $$A\subset\mathbb Z^3$$ be a set of positive upper Banach density. Is it true that there exists a set $$B\subset\mathbb Z$$ of positive upper density such that $$A-A\supset B\times B\times B ?$$
Question 2. Let $$A\subset\mathbb Z$$ be a set of positive upper Banach density. Is it true that there exists a set $$B\subset\mathbb Z$$ of positive upper density such that $$A-A\supset B+B+B ?$$
On the other hand one can prove the following facts.
Theorem 3.2. Let $$A\subset\mathbb Z^m$$ be a set of positive upper Banach density. Then there exists an infinite set $$B\subset\mathbb Z$$ such that $$A-A\supset B^m$$, where $$B^m$$ stands for the $$m$$-fold Cartesian product of $$B$$ with itself.
Theorem 5.3. Let $$A\subset\mathbb Z^m$$ be a set of positive upper Banach density and let $$v_1,v_2,\dots,v_n\in \mathbb Z^m.$$ Then there exists a sets $$B\subset\mathbb N$$ of positive upper density such that for any $$k_1, k_2, \dots, k_n\in B$$ the set $$A$$ contains a congruent image of the set $$\{0, k_1v_1, k_2v_2,\dots, k_nv_n\}.$$
Reviewer: V. Bergelson

### MSC:

 11B13 Additive bases, including sumsets 28D05 Measure-preserving transformations 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 11B05 Density, gaps, topology
Full Text: