Fourier-uniform sets with few 4-term arithmetic progressions

Fourier analysis is an important tool in additive combinatorics, famously used by Roth to find 3-term arithmetic progressions in dense subsets of the integers. One of the key properties used in his proof is that a Fourier-uniform set A consisting of aN elements of Z/NZ must contain approximately a^3N^2 3-term arithmetic progressions.

It is widely believed that Fourier analysis is not a sufficiently powerful tool to find more complicated structures such as 4-term arithmetic progressions. Specifically, Ruzsa conjectured that there exist Fourier-uniform sets consisting of aN elements of Z/NZ that contain fewer than a^CN^2 4-term arithmetic progressions for all positive C.

We show that Ruzsa's conjecture is equivalent to a seemingly-unrelated conjecture in arithmetic Ramsey theory: that there exist colorings of the first N integers using N^{o(1)} colors that have no symmetrically-colored 4-term arithmetic progressions (a progression is symmetrically-colored if its first and last term are given the same color and its middle terms are given the same color). We use this equivalence to give an example of a Fourier-uniform set with only a^{4.08}N^2 4-term arithmetic progressions, which is currently the best-known construction.

Based on joint work with MingYang Deng and Yufei Zhao.