Quotients of dense subsets of integers and short distances between product elements
type: Item
instance of: scholarly article
title: Quotients of dense subsets of integers and short distances between product elements
zbMATH Open document ID: 1490.11032
DOI: 10.1134/S000143462201014X
published in: Mathematical Notes
publication date: 2022-03-14T00:00:00Z
review text: By using results and methods proposed by \textit{J. Cilleruelo} et al. [Bull. Lond. Math. Soc. 42, No. 3, 517--526 (2010; Zbl 1205.11016)], the author refines a lower bound on the set of (Minkowski) quotients of two subsets \(A\) and \(B\) from finite intervals of natural numbers with positive density. As a consequence, he proves that if \(a\) and \(b\) are real numbers from \((0,1]\) , and if \(A\) and \(B\) are infinite subsets of natural numbers whose upper asymptotic densities are \(a, b\) respectively , and by putting \(d=2/ab\) , then there exist infinitely many pairs \((u,v)\in AB\) , \(u\ne v\) such that \(\lvert u-v\rvert\) =O(exp((c.log d/log log d)/ab)), c being an absolute constant, and \(AB\) denoting the Minkowski product of the sets \(A\) and \(B\).
reviewed by: József Sándor
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 7488486
zbMATH Keywords: divisibility
zbMATH Keywords: products
zbMATH Keywords: quotients
zbMATH Keywords: distances
author: Yu. N. Shteĭnikov
MaRDI profile type: Publication
cites work: Unsolved problems in number theory
cites work: On the distribution of products of members of a sequence with positive density
cites work: Q3436134
cites work: On a question of Sárközy on gaps of product sequences
cites work: The number of rational numbers determined by large sets of integers
cites work: On product sets of rationals
cites work: Quotient and product sets of thin subsets of the positive integers
cites work: On the size of the quotient of two subsets of positive integers
full work available at URL: s000143462201014x
OpenAlex ID: W4213445154
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description: scientific article; zbMATH DE number 7488486