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# Pair Correlation of Squares in $p$-Adic Fields

Published:2003-04-01
Printed: Apr 2003
• Alexandru Zaharescu
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## Abstract

Let $p$ be an odd prime number, $K$ a $p$-adic field of degree $r$ over $\mathbf{Q}_p$, $O$ the ring of integers in $K$, $B = \{\beta_1,\dots, \beta_r\}$ an integral basis of $K$ over $\mathbf{Q}_p$, $u$ a unit in $O$ and consider sets of the form $\mathcal{N}=\{n_1\beta_1+\cdots+n_r\beta_r: 1\leq n_j\leq N_j, 1\leq j\leq r\}$. We show under certain growth conditions that the pair correlation of $\{uz^2:z\in\mathcal{N}\}$ becomes Poissonian.
 MSC Classifications: 11S99 - None of the above, but in this section 11K06 - General theory of distribution modulo $1$ [See also 11J71] 1134 - unknown classification 1134