Sum-product estimates and finite point configurations over p-adic fields
<p>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 $\Q_p</p><p>d$ that will guarantee $E\cdot E+E\cdot E+\ldots+E\cdot E$ has positive Haar measure. Under a mild regularity assumption, we establis...
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| Language: | EN |
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University of Rochester
2017
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| Online Access: | http://pqdtopen.proquest.com/#viewpdf?dispub=10237020 |
| Summary: | <p>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 $\Q_p</p><p>d$ 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. |
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