Special Year Research Seminar

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb{Z}/N\mathbb{Z}$ 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 $\alpha^{4+c}$. 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 $N^{o(1)}$-color of $[N]$ avoiding symmetrically-colored 4-APs?

Joint work with Mingyang Deng and Jonathan Tidor.