Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates
| dc.creator | Chapman, Jeremy | |
| dc.creator | Erdogan, M. Burak | |
| dc.creator | Hart, Derrick | |
| dc.creator | Iosevich, Alex | |
| dc.creator | Koh, Doowon | |
| dc.date | 2009-03-25 | |
| dc.date | 2009-03-26 | |
| dc.date.accessioned | 2026-07-07T12:56:19Z | |
| dc.date.available | 2026-07-07T12:56:19Z | |
| dc.description | An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $α>0$ such that $|Δ(E)| \gtrsim q$ whenever $|E| \gtrsim q^α$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $Δ(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. 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 $Δ_y(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x\in E\}$ 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 $Δ_y(E)$ 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 $Π_y(E)=\{x\cdot y: x\in E\}$. 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 $α>0$ such that $E$ contains a congruent copy of every $k$ dimensional simplex whenever $|E| \gtrsim q^α$. Here the authors improve on known results (for $k>3$) using Fourier analytic methods, showing that $α$ may be taken to be $\frac{d+k}{2}$. | |
| dc.description | A few corrections | |
| dc.identifier | https://arxiv.org/abs/0903.4218 | |
| dc.identifier | http://arxiv.org/abs/0903.4218 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/224545 | |
| dc.subject | Combinatorics | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates | |
| dc.type | text |