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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
Subjects: Mathematics
Keywords: Combinatorics, Extremal problems, Harmonic analysis, Local fields, Nonarchimedean, P-adic
Publication Number: 10237020
ISBN: 978-1-369-46283-8
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