On a problem of Erdős and Szemerédi
type: Item
instance of: scholarly article
title: On a problem of Erdős and Szemerédi
zbMATH Open document ID: 0578.10057
DOI: 10.1016/0022-314X(86)90012-0
author: F. Grupp
author: Gerhard Bantle
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: \textit{P. Erdős} proved [Acta Arith. 12, 175-182 (1966; Zbl 0147.02601)] the following: Let \({\mathcal B}=\{b_ i\}^{\infty}_{i=1}\) be a sequence of distinct positive integers such that \((b_ i,b_ j)=1\) for \(i\neq j\) and the sum of \(1/b_ i\) converges. Then there exists \(\alpha\) such that \(0<\alpha <1\) and for \(x\geq x_ 0\) the interval \((x-x^{\alpha},x]\) contains integers divisible by no element of \textit{\({\mathcal B}\). E. Szemerédi} [ibid. 23, 359-361 (1973; Zbl 0266.10045)] showed that one can take \(\alpha =+\epsilon\). Now the authors make further progress in this interesting problem by showing that \(\alpha =9/20+\epsilon\) is permissible, estimating nontrivially certain exponential sums by using ideas due mainly to H. Iwaniec.
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 3924917
zbMATH Keywords: estimates of exponential sums
zbMATH Keywords: infinite sequence of integers
zbMATH Keywords: non- divisibility
zbMATH Keywords: sequence of distinct positive integers
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
cites work: Power mean values of the Riemann zeta‐function
cites work: An Additive Divisor Problem
cites work: On the difference of consecutive terms of sequences defined by divisibility properties
cites work: A new form of the error term in the linear sieve
cites work: \(P_ 2\) in short intervals
cites work: Contributions to the Erdös-Szemerédi theory of sieved integers
cites work: On the difference of consecutive terms of sequences defined by divisibility properties, II
full work available at URL: 0022-314x(86)90012-0
OpenAlex ID: W2003012675
description: scientific article; zbMATH DE number 3924917