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# Dissertation/Thesis Abstract

Sum-product estimates and finite point configurations over p-adic fields
by Ethier, Dillon, Ph.D., University of Rochester, 2016, 61; 10237020
Abstract (Summary)

We examine Erd\"{o}s-Falconer type problems in the setting of $p$-adic numbers, and establish bounds on the size of a set $E$ in $\Q_p d$ that will guarantee $E\cdot E+E\cdot E+\ldots+E\cdot E$ has positive Haar measure. Under a mild regularity assumption, we establish a lower bound on the dimension of a set that determines a set of simplices of positive measure, which reduces to an analogue of the distance problem when $1$-simplices are considered. Using the Mattila integral, we establish a different bound that improves upon the first bound when the dimension of the simplices is close to the ambient dimension.

Indexing (document details)
 Advisor: Iosevich, Alex, Quillen, Alice Commitee: Pakianathan, Jonathan, Rajeev, Sarada School: University of Rochester Department: School of Arts and Sciences School Location: United States -- New York Source: DAI-B 78/05(E), Dissertation Abstracts International Source Type: DISSERTATION Subjects: Mathematics Keywords: Combinatorics, Extremal problems, Harmonic analysis, Local fields, Nonarchimedean, P-adic Publication Number: 10237020 ISBN: 978-1-369-46283-8