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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
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