A concentration function estimate and intersective sets from matrices
type: Item
instance of: scholarly article
title: A concentration function estimate and intersective sets from matrices
zbMATH Open document ID: 1292.05062
author: Randall Mccutcheon
author: Paul Balister
published in: Israel Journal of Mathematics
publication date: 2012-11-13T00:00:00Z
review text: The content of the paper is related to ergodic methods in combinatorics and arithmetic, as initiated by \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)], and continues previous works, in particular of \textit{V. Bergelson} et al. [Topol. Proc. 23(Spring), 23--60 (1998; Zbl 0946.22002)]. A set \(R \subset \mathbb Z\) is said to be density-intersective if, for every \(A \subset \mathbb N\) with \[ \limsup_{b-a \to \infty} {|A \cap\{a+1, ...,b\}| \over b-a} >0, \] one has \(R \cap (A-A) \not = \emptyset\). Via the correspondence principle of Furstenberg, an equivalent formulation is that \(R\) is a set of measurable recurrence, i.e., if \((X, \mu, T)\) is a measure-preserving dynamical system, then for every measurable subset \(A\) with \(\mu(A) > 0\) there is an \(n\in R\) such that \(\mu(A \cap T^{-n} A) > 0\). A general conjecture is that, for every infinite integer matrix \((d_{i,j})\), the set \[ R= \sum_{i,j \in \alpha; \, i>j} d_{i,j}, \quad\alpha \subset \mathbb N, \quad |\alpha|< \infty, \] is a density-intersective set. This was proved in some cases. The special case \(d_{i,j} = 1, \forall i, j\), follows from a result of Sárközy (1978) on difference sets of sequences of integers. In this paper, the authors give sufficient conditions on \((d_{i,j})\) for the set \(R\) to be a density-intersective set. Several results are expressed in terms of \(p\)-limits where \(p\) is an ultrafilter. In order to state more precisely one of the results, let us introduce some notations and definitions. Let \(\mathcal F\) denote the set of non-empty finite subsets of \(\mathbb N\), \(\mathcal F(\{1, 2,\dots, n\})\) be the set of non-empty finite subsets of \(\{1, 2,\dots, n\}\). A subset \(\mathcal A\) of \(\mathcal F\) has positive upper density if \(\limsup_{n \to \infty} 2^{-n} | \mathcal A \cap \mathcal F(\{1, 2,\dots, n\})| > 0\). Recall that a subset \(S \subset \mathbb Z\) is syndetic if there is a finite set \(F \subset \mathbb Z\) such that…
reviewed by: Jean-Pierre Conze
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 15B35
Mathematics Subject Classification ID: 22A15
zbMATH DE Number: 6105479
zbMATH Keywords: density-intersective set
zbMATH Keywords: recurrence
zbMATH Keywords: syndetic set
zbMATH Keywords: ultrafilter
zbMATH Keywords: correspondence principle of Furstenberg
MaRDI profile type: Publication
full work available at URL: s11856-011-0176-4
OpenAlex ID: W2081813854
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DOI: 10.1007/S11856-011-0176-4
description: scientific article; zbMATH DE number 6105479