On large subsets of F<i>nq</i> with no three-termarithmetic progression

Journal article (2017)

Authors

Jordan S. Ellenberg University of Wisconsin-Madison
Dion Gijswijt Discrete Mathematics and Optimization - EEMCS
Research Group
Discrete Mathematics and Optimization (EEMCS) (TU Delft)
Copyright: © 2017 Jordan S. Ellenberg, Dion Gijswijt
DOI: https://doi.org/10.4007/annals.2017.185.1.8
Additive combinatorics Additive number theory Arithmetic progressions Cap sets
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Published Date

2017

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Discrete Mathematics and Optimization

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
. 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

.

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