Uniform sets with few progressions via colorings

TL;DR

This paper investigates the existence of Fourier-uniform subsets of cyclic groups with low 4-term arithmetic progression density, connecting combinatorial coloring problems to Fourier analysis and extending results to longer progressions and patterns.

Contribution

It establishes equivalences between Ruzsa's question on Fourier-uniform sets and coloring problems, and provides new constructions and bounds for uniform sets avoiding certain arithmetic progressions.

Findings

Ruzsa's question is linked to colorings without symmetrically colored 4-APs.

Constructs Fourier-uniform sets with low 4-AP density under certain conditions.

Extends results to all even-length APs and arbitrary one-dimensional patterns.

Abstract

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-AP) density at most $\alpha^C$, for arbitrarily large $C$. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c>0$. We show that an affirmative answer to Ruzsa's question would follow from the existence of an $N^{o(1)}$-coloring of $[N]$ without symmetrically colored 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa's question is equivalent to our arithmetic Ramsey question. We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and $k$-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalizations to arbitrary one-dimensional patterns.