Uniform Sets with Few Progressions via Colorings
Ruzsa asked whether there exist Fourier-uniform subsets of ℤ/Nℤ with very few 4-term arithmetic progressions (4-AP). The standard pedagogical example of a Fourier uniform set with a "wrong" density of 4-APs actually has 4-AP density much higher than random. Can it instead be much lower than random? Gowers constructed Fourier uniform sets with 4-AP density at most α4+c. It remains open whether a superpolynomial decay is possible. We will discuss this question and some variants. We relate it to an arithmetic Ramsey question: can one No(1)-color of [N] avoiding symmetrically-colored 4-APs?