小盒子上的双线性Kloosterman和与随机游走的一致性
Bilinear Kloosterman sums over small boxes and uniformity of a random walk
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中文总结 AI 辅助
该研究证明了有限域小盒子上双线性Kloosterman和的非平凡界，分析了基于该和构造的有限域随机游走的傅里叶系数衰减性，得到其分布收敛到均匀分布的结论及熵收敛速率界。
中文摘要 AI 辅助
给定任意有限域$\boldsymbol{\text{F}}_{p^n}$的加法特征$\boldsymbol{\text{\textpsi}}$及元素$a,b \boldsymbol{\text{\textin}} \boldsymbol{\text{F}}_{p^n}$且$b \neq 0$，我们证明双线性Kloosterman和$\boldsymbol{\text{\textsum}}_{x\boldsymbol{\text{\textin}} B_1}\boldsymbol{\text{\textsum}}_{y\boldsymbol{\text{\textin}} B_2}\boldsymbol{\text{\textpsi}}(axy+bx^{-1}y^{-1})$的界，其中$B_1,B_2$表示$\boldsymbol{\text{F}}_{p^n}$中的盒子；盒子是$\boldsymbol{\text{F}}_{p^n}$在$\boldsymbol{\text{F}}_p$上的固定基下，将域元素的系数限制在$\boldsymbol{\text{F}}_p$的区间内得到的坐标平行六面体。当$|B_1||B_2|>p^{n/2+\boldsymbol{\text{\textepsilon}}}$时，我们的估计是非平凡的，因此该范围是Weil界无法覆盖的。我们还考虑$\boldsymbol{\text{F}}_{p^n}$上的随机游走，其定义为$S_k=S_0+W_1+\boldsymbol{\text{\textcdots}}+W_k$，其中$W_i=aX_iY_i+b(X_iY_i)^{-1}$，$X_i,Y_i$分别是$B_1$和$B_2$上独立均匀分布的随机变量。我们证明$S_k$的非平凡傅里叶系数指数衰减，因此$S_k$的每个非零$\boldsymbol{\text{F}}_p$-线性投影及$S_k$的完整分布分别收敛到$\boldsymbol{\text{F}}_p$和$\boldsymbol{\text{F}}_{p^n}$上的均匀分布，同时我们还得到$S_k$的熵收敛到其最大值的速率界。
英文摘要
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.
