The analogue of Erdős-Turán conjecture in \(\mathbb Z_m\)
type: Item
instance of: scholarly article
title: The analogue of Erdős-Turán conjecture in \(\mathbb Z_m\)
zbMATH Open document ID: 1191.11006
published in: Journal of Number Theory
publication date: 2008-11-18T00:00:00Z
review text: The particular conjecture of Erdős and Turán to which the title of the paper refers is that if \(A \subset \mathbb N\) is such that the support of \(1_A \ast 1_A\) contains every sufficiently large integer then it must be unbounded. Although the conjecture itself is far from resolution some interesting partial results are know. Of special interest is the result of Ruzsa that the modular analogue is false: writing \(R_m\) for the smallest number such that there is a set \(A \subset \mathbb Z/m\mathbb Z\) with \(A+A=\mathbb Z/m\mathbb Z\) and \(\| 1_A \ast 1_A\| _\infty \leq R_m\), Ruzsa showed that \(R_m = O(1)\). It is the purpose of the paper under review to show that \(R_m \leq 288\). The argument itself rests on two key facts. First, the author provides a careful analysis of the case when \(m=2p^2\) for some prime \(p\). In this instance he is able to give examples of sets \(A\) with \(A+A=\mathbb Z/m\mathbb Z\) and \(\| 1_A \ast 1_A\| _\infty \leq 48\). This is then combined with the fact that if \(m_1 < m_2 \leq 3m_1/2\) then \(R_{m_1} \leq 6R_{m_2}\) to complete the proof. The approach is combinatorial and the paper also includes a clutch of related open questions, although perhaps its most novel selling point is the video summary which may be found on youtube.
reviewed by: Tom Sanders
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B34
zbMATH DE Number: 5368231
zbMATH Keywords: Erdős-Turán conjecture
zbMATH Keywords: additive bases
zbMATH Keywords: representation function
zbMATH Keywords: Ruzsa's number
author: Yonggao Chen
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.03.005
OpenAlex ID: W2033420558
Wikidata QID: Q123074766
cites work: On the multiplicative representation of integers
cites work: On a Problem of Sidon in Additive Number Theory, and on some Related Problems
cites work: Q5515488
cites work: Unique representation bases for the integers
cites work: Representation functions of additive bases for abelian semigroups
cites work: Inequalities concerning the function π(x): Applications
cites work: On multiplicative bases in abelian groups
cites work: A just basis
cites work: A basis of Zm
cites work: A basis of Zm, II
DOI: 10.1016/J.JNT.2008.03.005
Recommended article: An analogue of the Erdős-Ginzburg-Ziv theorem over \(\mathbb{Z}\)
Recommended article: Q3628988
Recommended article: The Erdős–Turán Conjecture in the positive rational numbers
Recommended article: Analytic Erdős-Turán conjectures and Erdős-Fuchs theorem
Recommended article: Q4882474
Recommended article: Q4416548
Recommended article: On a Conjecture of Erdős, Gallai, and Tuza
Recommended article: The Erdős-Szekeres theorem and congruences
Recommended article: On the Erdős-Turán conjecture
description: scientific article; zbMATH DE number 5368231