## Abstract

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 language | English |
---|---|

Pages (from-to) | 339-343 |

Number of pages | 5 |

Journal | Annals of Mathematics |

Volume | 185 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

### Bibliographical note

Accepted author manuscript## Keywords

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

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