A concentration function estimate and intersective sets from matrices
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 \(\mathbb Z = S + F\). One of the many results of the paper is the following (Corollary 4.12): Let \((c_{n,j})_{i>j}\) be an infinite, lower triangular, natural-number-valued matrix. Suppose that for every \(j \in \mathbb N\), \(c_{n,j} = o(n/\log n)\) as \(n \to \infty\). Then the function \(\alpha \to v(\alpha) := \sum_{j \in \alpha} c_{n,j}\) on \(\mathcal F\) is ``covering''. This means that for some \(k \in \mathbb N\), for every \(\mathcal A \subset \mathcal F\) with positive upper density, the set \(kv(\mathcal A) - kv(\mathcal A)\) is syndetic. As shown in the paper, this property implies in particular that \(v(\mathcal F)\) is a set of measurable recurrence, hence density-intersective. The proofs rely on techniques of ultrafilters and a version of the Van der Corput Lemma, as well as the following concentration function estimate, which is of independent interest (Lemma 4.10): There exist positive constants \(c\), \(C\) having the following properties. Suppose \(N \in \mathbb N\) and \((d_n)_{n=1}^N\) is a sequence of integers with \(d_1 = 1\) and \(1 \leq d_n \leq \max\{1, c \sqrt{n/\log n}\}\) for \(n \geq 2\). If \((X_n)_{n=1}^N\) are independent random variables with \(\mathbb P(X_i = 0) = \frac12 = \mathbb P(X_i = d_i)\), then \(\mathbb P(\sum_{n=1}^N X_n = k) \leq C( \sum_{n=1}^N d_n^2)^{-\frac12}\) for all \(k\).
scientific article; zbMATH DE number 6105479