Uniform sets with few progressions via colourings

Ruzsa asked whether there exist Fourier-uniform subsets Z/NZ of with density α and 4-term arithmetic progression (4-AP) density at most α^C, for arbitrarily large C. Gowers constructed Fourier uniform sets with density α and 4-AP density at most α^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 N⁰⁽¹⁾-colouring of [N] without symmetrically coloured 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of Z/NZ, 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≥5, we show that there exist U^k-2-uniform subsets of Z/ZN with density α and k-AP density at most α^{ck log(1/α)}. We also prove generalisations to arbitrary one-dimensional patterns.