Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates

Details Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates

2009-03-25

arxiv.org/abs/0903.4218v2

Mathematics

Combinatorics

A few corrections

Scientific paper

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $\alpha>0$ such that $|\Delta(E)| \gtrsim q$ whenever $|E| \gtrsim q^{\alpha}$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $\Delta(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. In two dimensions we improve the known exponent to $\tfrac{4}{3}$, consistent with the corresponding exponent in Euclidean space obtained by Wolff. The pinned distance set $\Delta_y(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x\in E\}$ for a pin $y\in E$ has been studied in the Euclidean setting. Peres and Schlag showed that if the Hausdorff dimension of a set $E$ is greater than $\tfrac{d+1}{2}$ then the Lebesgue measure of $\Delta_y(E)$ is positive for almost every pin $y$. In this paper we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set $\Pi_y(E)=\{x\cdot y: x\in E\}$. Under the additional assumption that the set $E$ has cartesian product structure we improve the pinned threshold for both distances and dot products to $\frac{d^2}{2d-1}$. A generalization of the Falconer distance problem is determine the minimal $\alpha>0$ such that $E$ contains a congruent copy of every $k$ dimensional simplex whenever $|E| \gtrsim q^{\alpha}$. Here the authors improve on known results (for $k>3$) using Fourier analytic methods, showing that $\alpha$ may be taken to be $\frac{d+k}{2}$.

Affiliated with

Also associated with

No associations

Rating

Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-67795