On large subsets of $F_q^n$ with no three-term arithmetic progression

Jordan S. Ellenberg; Dion Gijswijt

On large subsets of $F_q^n$ with no three-term arithmetic progression

Combinatorics 2016-05-31 v1 Number Theory

Authors: Jordan S. Ellenberg , Dion Gijswijt

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 $F_q^n$ with no three terms in arithmetic progression by $c^n$ with $c < q$ . For $q=3$ , the problem of finding the largest subset with no three terms in arithmetic progression is called the `cap problem'. Previously the best known upper bound for the cap problem, due to Bateman and Katz, was $O(3^n / n^{1+\epsilon})$ .

Keywords

f_q^n subset three-term arithmetic progression cap problem croot-lev-pach method upper bound finite field combinatorics

Cite

@article{arxiv.1605.09223,
  title  = {On large subsets of $F_q^n$ with no three-term arithmetic progression},
  author = {Jordan S. Ellenberg and Dion Gijswijt},
  journal= {arXiv preprint arXiv:1605.09223},
  year   = {2016}
}

Comments

4 pages. This paper supersedes arXiv:1605.05492 and combines the solutions to the cap set problem independently obtained by the two authors

On large subsets of $F_q^n$ with no three-term arithmetic progression

Abstract

Keywords

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