On large subsets of Fnq with no three-termarithmetic progression

Jordan S. Ellenberg, Dion Gijswijt

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In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of F n q  Fqn with no three terms in arithmetic progression by c n  cn with c<q c<q . For q=3 q=3 , the problem of finding the largest subset of F n 3  F3n with no three terms in arithmetic progression is called the cap set problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz, was on order n −1−ϵ 3 n  n−1−ϵ3n .
Original languageEnglish
Pages (from-to)339-343
Number of pages5
JournalAnnals of Mathematics
Issue number1
Publication statusPublished - 2017

Bibliographical note

Accepted author manuscript


  • additive combinatorics
  • additive number theory
  • arithmetic progressions
  • cap sets


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