  Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb
Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-APs)
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.
