Roth's theorem in Z_4^n

TL;DR

This paper proves that large subsets of Z_4^n without three-term arithmetic progressions are extremely rare, with their size growing slower than 4^n/n, extending Roth's theorem to this setting.

Contribution

It establishes a new bound on the size of progression-free subsets in Z_4^n, advancing understanding of additive combinatorics in this group.

Findings

Progression-free subsets are of size o(4^n/n)

Extends Roth's theorem to Z_4^n

Provides bounds on the density of such subsets

Abstract

We show that if A is a subset of Z_4^n containing no three-term arithmetic progression in which all the elements are distinct then |A|=o(4^n/n).