Arithmetic progressions with a pseudorandom step

Elad Aigner-Horev, Hiêp Hàn

Research output: Contribution to journalArticlepeer-review


Let α, σ > 0 and let A and S be subsets of a finite abelian group G of densities α and σ, respectively, both independent of |G|. Without any additional restrictions, the set A need not contain a 3-term arithmetic progression whose common gap lies in S. What is then the weakest pseudorandomness assumption that if put on S would imply that A contains such a pattern?More precisely, what is the least integer k≥2 for which there exists an η=η(α, σ) such that ‖S-σ‖Uk(G)≤η implies that A contains a non-trivial 3-term arithmetic progression with a common gap in S? Here, ‖{dot operator}‖Uk(G) denotes the kth Gowers norm.For G=Zn we observe that k must be at least 3. However for G=Fnp we show that k= 2 is sufficient, where here p is an odd prime and n is sufficiently large.

Original languageEnglish
Pages (from-to)447-455
Number of pages9
JournalElectronic Notes in Discrete Mathematics
StatePublished - Nov 2015


  • Arithmetic progressions
  • Pseudorandom sets
  • Regularity method


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