Bilinear Kloosterman sums over small boxes and uniformity of a random walk

Abstract

Given an additive character $ψ$ of an arbitrary finite field $\mathbb{F}_{p^n}$ and elements $a,b\in \mathbb{F}_{p^n}$, $b\neq 0$, we prove a bound on bilinear Kloosterman sums $\sum_{x\in B_1}\sum_{y\in B_2}ψ(axy+bx^{-1}y^{-1})$, where $B_1, B_2$ denote boxes in $\mathbb{F}_{p^n}$. Here, a box is a coordinate parallelepiped obtained by restricting the coefficients of field elements, with respect to a fixed basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$, to intervals in $\mathbb{F}_p$. Our estimates are nontrivial when $|B_1||B_2|>p^{n/2+ε}$ and hence in a range not accessible by the Weil bound. We also consider the random walk on $\mathbb{F}_{p^n}$ defined by $S_k=S_0+W_1+\cdots+W_k$, where $W_i=aX_iY_i+b(X_iY_i)^{-1}$ and $X_i,Y_i$ are independent uniformly distributed random variables on $B_1$ and $B_2$ respectively. We show that the nontrivial Fourier coefficients of $S_k$ decay exponentially. Consequently, every nonzero $\mathbb{F}_p$-linear projection of $S_k$, as well as the full distribution of $S_k$ converge to the uniform distribution on $\mathbb{F}_p$ and $\mathbb{F}_{p^n}$, respectively. We also obtain bounds on the rate at which the entropy of $S_k$ converges to its maximal value.