On a problem of erdős and turán and some related results

Noga Alon, Mihail N. Kolountzakis

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

We employ the probabilistic method to prove a stronger version of a result of Helm, related to a conjecture of Erdo{combining double acute accent}s and Turán about additive bases of the positive integers. We show that for a class of random sequences of positive integers A, which satisfy ∣A ∩ [1, x]∣ ≫ √x with probability 1, all integers in any interval [ I, N] can be written in at least c1 log N and at most c2 log N ways as a difference of elements of A ∩ [1, N2]. We also prove several results related to another result of Helm. We show that for every sequence of positive integers M, with counting function M(x), there is always another sequence of positive integers A such that M ∩ (A - A) = ∅ and A(x) > x/(M(x) + 1). We also show that this result is essentially best possible, and we show how to construct a sequence A with A(x) > cx/(M(x) + 1) for which every element of M is represented exactly as many times as we wish as a difference of elements of A.

Original languageEnglish (US)
Pages (from-to)82-93
Number of pages12
JournalJournal of Number Theory
Volume55
Issue number1
DOIs
StatePublished - Nov 1995
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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