Uniform dilations
type: Item
instance of: scholarly article
title: Uniform dilations
zbMATH Open document ID: 0756.11020
DOI: 10.1007/BF01895704
author: Yuval Peres
author: Noga Alon
published in: Geometric and Functional Analysis. GAFA
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/58132
review text: Generalizing a result of \textit{S. Glasner} [Isr. J. Math. 32, 161-172 (1979; Zbl 0406.54023)], \textit{D. Berend} and \textit{Y. Peres} [J. Lond. Math. Soc., II. Ser., to appear (Zbl 0728.11037)], proved that every sufficiently large finite subset \(X\) of \([0,1]\) (\(\text{card} X\geq k(\varepsilon)\geq c/\varepsilon^ 2\)) has a dilation \(nX\bmod 1\) with small maximal gap (\(nX\) is \(\varepsilon\)-dense). In this interesting paper the authors give quantitative versions of this result which are almost optimal. (\(k(\varepsilon)\leq 1/\varepsilon^{2+\gamma}\), \(\gamma>0\) arbitrary, \(\varepsilon\leq\varepsilon_ 0(\gamma)\), Theorem 1.1.) (Analogous generalizations hold if \(n\in\mathbb{N}\) is replaced by \(n\in S\subset\mathbb{N}\), with positive upper Banach density.) Furthermore the sequence \(nX\) (if \(X\) consists of \(k\) different points) has small discrepancy (\(\leq k^{\gamma-1/2}\), \(k>k_ 0(\gamma)\), Theorem 1.2). The proof is based on a second moment argument. (Remark to Proposition 2.1: \(\text{Prob}[Y^ I=0]=(1-\delta)^ k<1/k\delta\)) which reduces the problem to an estimate on the number of edges in a certain graph. Cycles in this graph correspond to solutions of diophantine equations: The growth asymptotics of these solutions, which can be determined from properties of lattices in Euclidean space, yield the required estimate. There are interesting connections with the distribution of quadratic residues modulo \(p\), considering the according problem in the cyclic group \(\mathbb{Z}_ p\). Smaller gaps may be obtained by considering images \(f(X)\) under polynomials of higher degree. Proposition 1.3 can also be proved using harmonic analysis [\textit{J. Bourgain}, Isr. J. Math. 67, 291- 311 (1989; Zbl 0692.43005)]. The paper also contains a proof of Theorem 1.1 in a slightly sharper form (again using harmonic analysis, as proposed by L. Carleson). It is even possible to replace \(n\) by a prime \(p\) or by \(f(n)\), \(f\) a non constant polynom…
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11J13
zbMATH DE Number: 56871
zbMATH Keywords: second moment method
zbMATH Keywords: density mod 1
zbMATH Keywords: dilation
zbMATH Keywords: small maximal gap
zbMATH Keywords: positive upper Banach density
zbMATH Keywords: small discrepancy
reviewed by: Harald Rindler
MaRDI profile type: Publication
cites work: A fast and simple randomized parallel algorithm for the maximal independent set problem
cites work: Cube Slicing in R n
cites work: Asymptotically Dense Dilations of Sets on the Circle
cites work: On \(\Lambda\) (p)-subsets of squares
cites work: On Character Sums and Primitive Roots†
cites work: A Note on the Distribution of Residues and Non-Residues
cites work: Q3900124
cites work: Q5610469
cites work: Almost periodic sets and measures on the torus
cites work: Q3941473
cites work: Q4122353
cites work: Arithmetical properties of the digits of the multiples of an irrational number
cites work: Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height
cites work: Q5827024
cites work: Q5750924
OpenAlex ID: W4238741629
Recommended article: On uniformly distributed dilates of finite integer sequences
Recommended article: Equal sums in random sets and the concentration of divisors
Recommended article: Squares in sumsets
Recommended article: On the Normal Concentration of Divisors
Recommended article: On LeVeque's theorem about the uniform distribution (mod 1) of \((a_ j\,\cos \,a_ jx)^{\infty}_{j=1}\)
Recommended article: Quotient and product sets of thin subsets of the positive integers
Recommended article: On a class of uniformly distributed sequences of point sets
Recommended article: Q4500966
Recommended article: Six Standard Deviations Suffice
Recommended article: Point sets with distinct distances
description: scientific article; zbMATH DE number 56871