Generalization of a theorem of Erdős and Rényi on Sidon sequences

Javier Cilleruelo, Sándor Z. Kiss, Imre Z. Ruzsa, Carlos Vinuesa

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Erdo{double acute}s and Rényi claimed and Vu proved that for all h ≥ 2 and for all h > 0, there exists g = gh(ε) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ≫ x1/h-ε.We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a g that satisfies gh(ε) ≪ ε-1, improving the bound gh(ε) ≪ ε-h+1 obtained by Vu.Finally we use the "alteration method" to get a better bound for g3(ε), obtaining a more precise estimate for the growth of B3[g] sequences.

Original languageEnglish
Pages (from-to)455-464
Number of pages10
JournalRandom Structures and Algorithms
Volume37
Issue number4
DOIs
Publication statusPublished - Dec 8 2010

Keywords

  • Sidon sequences
  • The probabilistic method

ASJC Scopus subject areas

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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