Progression-free sets in Z_4^n are exponentially small

TL;DR

This paper proves that subsets of Z_4^n avoiding three-term arithmetic progressions are exponentially small relative to the size of the entire space, with a specific exponential bound.

Contribution

It establishes a new exponential upper bound on the size of progression-free sets in Z_4^n, improving understanding of their structure.

Findings

Progression-free sets in Z_4^n are exponentially small.

The size of such sets is less than 4^{0.926 n}.

Provides a quantitative bound on progression-free subsets.

Abstract

We show that for integer $n>0$, any subset $A \subset Z_4^n$ free of three-term arithmetic progressions has size $|A| < 4^{c n}$, with an absolute constant $c \approx 0.926$.