The Green-Tao theorem states that there exist arbitrarily long arithmetic progressions of prime numbers. Later works of Green, Tao and Ziegler provided an asymptotic for the number of progressions of length k inside the primes smaller than x. One of the crucial ingredient for progressions of length 4 is the quadratic uniformity of the moebius function, that is, roughly speaking, the fact that the Möbius function does not correlate with a quadratic phase.
A function field analogue would say that there exist affine subspaces of arbitrarily large dimensions. Such a statement was proven by Thai Hoang Lê en 2009, but without asymptotic. Here again, to get an asymptotic, one would need to understand correlations of the Möbius function with linear and quadratic phases. I will review joint work with Hoang on this question; we solved the linear case, and our method for the quadratic case relies on a conjectured quantitative improvement of a new additive-combinatorial structural result of our own which we called bilinear Bogolyubov theorem.