Density inequalities for sets of multiples
type: Item
instance of: scholarly article
title: Density inequalities for sets of multiples
zbMATH Open document ID: 0849.11013
author: Levon H. Khachatrian
author: Rudolf Ahlswede
published in: Journal of Number Theory
publication date: 1996-11-19T00:00:00Z
review text: For a finite set \(A\) of integers let \(M(A)\) be the set of natural numbers divisible by at least one element of \(A\), and let \({\mathbf d}\) denote asymptotic density. For two sets \(A\), \(B\) write \[ A\times B= \{ab\}, \qquad [A, B]= \{[a, b]\}, \qquad (A, B)= \{(a, b)\}, \] where \(a\in A\), \(b\in B\). A classical theorem of \textit{F. A. Behrend} [Bull. Am. Math. Soc. 54, 681-684 (1948; Zbl 0031.34602)] can be reformulated as \({\mathbf d} M[A, B]\geq {\mathbf d} M(A) {\mathbf d} M(B)\). Here this inequality is sharpened to \({\mathbf d} M(A, B){\mathbf d} M[A, B]\geq {\mathbf d} M(A) {\mathbf d} M(B)\), and the new inequality \[ {\mathbf d} M(A\times B)\leq {\mathbf d} M(A) {\mathbf d} M(B) \] is established.
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 844629
zbMATH Keywords: sets of multiples
zbMATH Keywords: asymptotic density
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
OpenAlex ID: W1997850074
DOI: 10.1006/JNTH.1995.1135
description: scientific article; zbMATH DE number 844629