Uniform sets with few progressions via colourings

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 $c \gt 0$ . We show that an affirmative answer to Ruzsa’s question would follow from the existence of an $N^{o(1)}$ -colouring of [ N ] without symmetrically coloured 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 generalisations to arbitrary one-dimensional patterns.