Uniform sets with few progressions via colorings

Abstract

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb{Z}/N\mathbb{Z}$ with density $\hat{I}\pm$ and 4-term arithmetic progression (4-AP) density at most $\hat{I}{\pm }^C$, for arbitrarily large $C$. Gowers constructed Fourier uniform sets with density $\hat{I}\pm$ and 4-AP density at most $\hat{I}{\pm }^{4+c}$ 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 $k\ge 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb{Z}/N\mathbb{Z}$ with density $\hat{I}\pm$ and $k$-AP density at most $\hat{I}{\pm }^{c_k\log (1/\hat{I}\pm )}$. We also prove generalizations to arbitrary one-dimensional patterns.

20 pages; typos corrected