We examine correlations of the Möbius function over Fq [t] with linear or quadratic phases, that is, averages of the formfor an additive character χ over Fq and a polynomial Q ∈ Fq[x0, . . . , xn−1] of degree at most 2 in the coefficients x0, . . . , xn−1 of f = i 0 if Q is linear and O q −n c for some absolute constant c > 0 if Q is quadratic.The latter bound may be reduced to O(q −c ′ n ) for some c ′ > 0 when Q(f ) is a linear form in the coefficients of f 2 , that is, a Hankel quadratic form, whereas for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.
The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set P in a Cartesian square F n p × F n p . More precisely, we perform horizontal and vertical sums and differences on P , that is, operations on the second coordinate when the first one is fixed, or vice versa. The structure we find is the zero set of a family of bilinear forms on a Cartesian product of vector subspaces. The codimensions of the subspaces and the number of bilinear forms involved are bounded by a function c(δ) of the density δ = |P | /p 2n only. The proof uses various tools of additive combinatorics, such as the (linear) Bogolyubov theorem, the density increment method, as well as the Balog-Szemerédi-Gowers and Freiman-Ruzsa theorems. P.-Y. Bienvenu, Institut Camille-Jordan, Université Lyon 1, 43 boulevard du 11 novembre 1918 69622
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