On the union of arithmetic progressions

Shoni Gilboa, Rom Pinchasi

Research output: Contribution to journalArticlepeer-review


We show that for any integer n ≥ 1 and real ε > 0, the union of n arithmetic progressions with pairwise distinct differences, each of length n, contains at least c (ε)n2-ε elements, where c (ε) is a positive constant depending only on ε. This estimate is sharp in the sense that the assertion becomes invalid for ε = 0. We also obtain estimates for the "asymmetric case" where the number of progressions is distinct from their lengths.

Original languageEnglish
Pages (from-to)1062-1073
Number of pages12
JournalSIAM Journal on Discrete Mathematics
Issue number3
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© 2014 Society for Industrial and Applied Mathematics.


  • Additive combinatorics
  • Arithmetic progressions
  • Bondy-Simonovits theorem


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