On Sums, Products, and the multidimensional Falconer problem

Erdos-Falconer distance problem, exponential sums, and Fourier analytic approach to incidence theorems in vector spaces over finite fields

We prove several incidence theorems in vector spaces over finite fields using bounds for various classes of exponential sums and apply these to Erdos-Falconer type distance problems.

Erdős–Falconer距离问题在有限域上的Delsarte线性规划方法

A Delsarte Linear Programming Approach to the Erdős--Falconer Distance Problem over Finite Fields

引入Delsarte线性规划方法，证明对偶维有限域上非退化二次型，当点集大小≥C q^{n/2+1/3}时，距离集包含正比例元素，改进了Iosevich–Rudnev的指数(n+1)/2至n/2+1/3。

Erdos distance problem in vector spaces over finite fields

We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let ${\Bbb F}_q$ be a finite field with $q$ elements and take $E \subset {\Bbb F}^d_q$, $d \ge 2$. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in ${\Bbb F}^d_q$ to provide estimates for minimum cardinality of the distance set $Δ(E)$ in terms of the cardinality of $E$. Kloosterman sums play an important role in the proof.

Littlewood's problem for sets with multidimensional structure

We give $L^1$-norm estimates for exponential sums of a finite sets $A$ consisting of integers or lattice points. Under the assumption that $A$ possesses sufficient multidimensional structure, our estimates are stronger than those of McGehee-Pigno-Smith and Konyagin. These theorems improve upon past work of Petridis.

Multidimensional Divide-and-Conquer and Weighted Digital Sums

This paper studies three types of functions arising separately in the analysis of algorithms that we analyze exactly using similar Mellin transform techniques. The first is the solution to a Multidimensional Divide-and-Conquer (MDC) recurrence that arises when solving problems on points in $d$-dimensional space. The second involves weighted digital sums. Write $n$ in its binary representation $n=(b_i b_{i-1}... b_1 b_0)_2$ and set $S_M(n) = \sum_{t=0}^i t^{\bar{M}} b_t 2^t$. We analyze the average $TS_M(n) = \frac{1}{n}\sum_{j<n} S_M(j)$. The third is a different variant of weighted digital su…

Eigenfunctions of the Multidimensional Linear Noise Fokker-Planck Operator via Ladder Operators

The eigenfunctions and eigenvalues of the Fokker-Planck operator with linear drift and constant diffusion are required for expanding time-dependent solutions and for evaluating our recent perturbation expansion for probability densities governed by a nonlinear master equation. Although well-known in one dimension, for multiple dimensions the eigenfunctions are not explicitly given in the literature. We develop raising and lowering operators for the Fokker-Planck (FP) operator and its adjoint, and use them to obtain expressions for the corresponding eigenvalues and eigenfunctions. We show that …