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Powers of sequences and recurrence. (English) Zbl 1173.37006

The authors study intersective subsets of integers. In order to state the results, we recall the following notions and terminologies. A set \(\Lambda\) of integers is said to have positive upper density \({\bar d}(\Lambda)\) if \({\bar d}(\Lambda) = \lim\sup_{N\to \infty} |\Lambda \cap \{-N, \ldots, N\}|/(2 N + 1) > 0\) and a set of integers \(R\) is called intersective if for every set of integers \(\Lambda\) with positive upper density, the equation \(x - y = r\) is solvable in \(x, y \in \Lambda\) and \(r\in\mathbb R\setminus \{0\}.\) This condition is equivalent to the property that \(\Lambda \cap (\Lambda - r) \neq \emptyset\) for some non-zero \(r \in\mathbb R\).
A well-known theorem of A. Sárközy [Acta Math. Acad. Sci. Hung. 31, 125–149 (1978; Zbl 0387.10033)] says that for any fixed positive integer \(k,\) the set of \(k\)th powers of integers is intersective. To the other direction, [V. Bergelson and I. J. Hø{a}land, Convergence in ergodic theory and probability. Papers from the conference, Ohio State University, Columbus, OH, USA, June 23–26, 1993. Berlin: de Gruyter. Ohio State Univ. Math. Res. Inst. Publ. 5, 91–110 (1996; Zbl 0958.28014)] constructed a set of integers that is not intersective but the set of squares of its elements is intersective. In this paper, the authors generalize this type of results. The first main result (Theorem A) of this paper is the following. Let \(G\) be a set of positive integers. Then there exists a set of integers \(R = \{r_1, r_2, \dots \}\) such that: for \(k \in\mathbb N\), the set \(R^k = \{r_1^k, r_2^k, \dots \}\) is intersective if and only if \(k \in G.\) In another main theorem (Theorem B), the authors generalize their result further to the case of \(\ell\)-intersective set. Here, a set of integers \(R\) is called \(\ell\)-intersective if for every set \(\Lambda\) of positive upper density contains an \(\ell + 1\) long arithmetic progression with a common difference from \(R \setminus \{0\}.\) The condition for \(\ell\)-intersective is equivalent to that \(\Lambda \cap (\Lambda - r) \cap \ldots \cap (\Lambda - \ell r) \neq \emptyset\) for some non-zero \(r\in R.\) Then, the second main result of this paper is as follows: Let \(\ell\) be a positive integer and \(G\) be a set of positive integers. There exists a set of integers \(R = \{r_1, r_2, \ldots \}\) such that: for \(k \in\mathbb N\), the set \(R^k = \{r_1^k, r_2^k, \ldots\}\) is \(\ell\)-intersective if and only if \(k \in G\).
To prove the main theorems, the authors interpret the notion of intersective sets as well as \(\ell\)-intersective sets in terms of ergodic theoretic language. Then, the above two theorems are reformulated from ergodic theoretical point of views. This enables the authors to apply several ergodic results to prove their main theorems.

MSC:

37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
28D05 Measure-preserving transformations
05D10 Ramsey theory
11B25 Arithmetic progressions
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