Uniform dilations
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 polynomial with integer coefficients.
scientific article; zbMATH DE number 56871
scientific article