Uniform sets with few progressions via colorings

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ 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 $\mathbb{Z}\mathrm{/}N\mathbb{Z}$, 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 $\mathbb{Z}\mathrm{/}N\mathbb{Z}$ with density $\alpha$ and $k$-AP density at most ${\alpha }^{c_k\log (1\mathrm{/}\alpha )}$. We also prove generalizations to arbitrary one-dimensional patterns.