On the density of integral sets with missing differences from sets related to arithmetic progressions
type: Item
instance of: scholarly article
title: On the density of integral sets with missing differences from sets related to arithmetic progressions
zbMATH Open document ID: 1229.11044
author: Ram Krishna Pandey
author: Amitabha Tripathi
published in: Journal of Number Theory
publication date: 2011-03-03T00:00:00Z
review text: Given a set \(M\) of positive integer, a set \(S\) of non-negative integers is said to be an \(M\)-set if \(a\in S\) and \(b\in S\) imply \(a-b\notin M\). In an unpublished problem collection T. S. Motzkin asks for determining the quantity \(\mu(M)=\bar{\delta}(S)\) where \(S\) varies over all \(M\)-sets and \(\bar{\delta}(S)\) stands for the upper asymptotic density. In the case \(|M|\leq2\) the problem was completely solved by \textit{D. G. Cantor} and \textit{B. Gordon} [J. Comb. Theory, Ser. A 14, 281--287 (1973; Zbl 0277.10043)], and there are known partial answers for several families of \(M\) with \(|M|\geq 3\). In the present paper some cases when \(M\) either contains an arithmetic progression or is contained in an arithmetic progression are settled.
reviewed by: Štefan Porubský
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 11B25
zbMATH DE Number: 5859970
zbMATH Keywords: Motzkin problem
zbMATH Keywords: upper asymptotic density
zbMATH Keywords: coloring
zbMATH Keywords: arithmetic sequence
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.09.013
OpenAlex ID: W2076935870
cites work: Flows, view obstructions, and the lonely runner
cites work: The lonely runner with seven runners
cites work: Sequences of integers with missing differences
cites work: Circular chromatic numbers and fractional chromatic numbers of distance graphs
cites work: View-obstruction problems in \(n\)-dimensional geometry
cites work: Colouring the real line
cites work: The channel assignment problem for mutually adjacent sites
cites work: Density of M-sets in arithmetic progression
cites work: Sets of integers with missing differences
cites work: Coloring of integer distance graphs
cites work: Coloring of distance graphs with intervals as distance sets
cites work: Fractional chromatic number and circular chromatic number for distance graphs with large clique size
cites work: Fractional chromatic number of distance graphs generated by two-interval sets
cites work: An Asymptotic Approach to the Channel Assignment Problem
cites work: Zwei Sätze über inhomogene diophantische Approximation von Irrationalzahlen
cites work: Q5475526
DOI: 10.1016/J.JNT.2010.09.013
description: scientific article; zbMATH DE number 5859970