Asymptotic upper bounds on progression-free sets in ${\mathbb{Z}}_p^n$

Abstract

We show that any subset of ${\mathbb{Z}}_p^n$ ($p$ an odd prime) without $3$-term arithmetic progression has size $O(p^{cn})$, where $c:=1-\frac{1}{18\log p}<1$. In particular, we find an upper bound of $O(2.84^n)$ on the maximum size of an affine cap in $GF(3{)}^n$.

This paper is superseded by a joint paper with Jordan Ellenberg: arXiv:1605.09223

References in corpus (2)

• Progression-free sets in Z_4^n are exponentially small
• On large subsets of $F_q^n$ with no three-term arithmetic progression

Cited by in corpus (3)

• On cap sets and the group-theoretic approach to matrix multiplication
• A nearly tight upper bound on tri-colored sum-free sets in characteristic 2
• Sarkozy's theorem in function fields