Linear and Quadratic Uniformity of the Möbius Function Over

Abstract

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<n xit i . Like in the integers, it is reasonable to expect that, due to the random-like behaviour of µ, such sums should exhibit considerable cancellation. In this paper we show that the correlation (1) is bounded by Oǫ q (− 1 4 +ǫ)n for any ǫ > 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.

Endomorphisms of positive characteristic tori: entropy and zeta function

“…The analytic number theory, more specifically the theory of characters and Lfunctions, on T F has been studied since at least 1965 in work of Hayes [Hay65]. Some relatively recent results include work of Liu-Wooley [LW10] on Waring's problem and the circle method in function fields and work of Porritt [Por18] and Bienvenu-Lê [BL19] on correlation between the Möbius function and a character over Z F . For a recent work in the ergodic theory side, we refer the readers to the paper by Bergelson-Leibman [BL16] and its reference in which the authors establish a Weyl-type equidistribution theorem.…”

A bilinear Bogolyubov theorem

“…The authors exploit this corollary, with the conjectured bound on c(δ) in Theorem 2, in a companion paper [1].…”

“…If P is a Cartesian product A×B for some subsets A, B ⊂ V , then using Theorem 1 once on each coordinate, we obtain a product A ′ × B ′ of subspaces of codimension O(log 4 δ −1 ). Also it is easy to see that c(δ) ≫ log δ −1 by considering a set such as the right-hand side of equation (1). Conjecture 3 says that, like in the linear case, this lower bound on δ should not be too far off the truth.…”

“…For a subset P ⊂ V × V , we define vertical and horizontal additive operations on P as follows: P V ± P = {(x, y 1 ± y 2 ) | ((x, y 1 ), (x, y 2 )) ∈ P 2 } Date: October 15, 2018. 1 and P H ± P = {(x 1 ± x 2 , y) | ((x 1 , y), (x 2 , y)) ∈ P 2 } where V and H mean vertical and horizontal. Note that V is also the name of the ambient space, but this should not create any confusion.…”

“…καp −6r |A|, which implies that there exists a linear form φ ∈ V * \ {0} such that for1 Polynomial under the polynomial Freiman-Ruzsa conjecture.…”