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Pair Correlation of Squares in $p$ -Adic Fields

Published online by Cambridge University Press:  20 November 2018

Alexandru Zaharescu
Alexandru Zaharescu*
Affiliation:
Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, Illinois 61801 USA., e-mail: zaharesc@math.uiuc.edu
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Abstract

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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 $K,\,B\,=\,\{{{\beta }_{1}},\ldots .{{\beta }_{r}}\}$ an integral basis of $K$ over ${{\mathbf{Q}}_{p}}$ , $u$ a unit in $O$ and consider sets of the form $N\,=\,\{{{n}_{1}}{{\beta }_{1}}\,+\ldots +\,{{n}_{r}}{{\beta }_{r}}\,:\,1\,\le \,{{n}_{j}}\,\le \,{{N}_{j}},\,1\,\le \,j\,\le \,r\}$ . We show under certain growth conditions that the pair correlation of $\{u{{z}^{2}}\,:\,z\,\in N\}$ becomes Poissonian.

Keywords

11S9911K06

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 2 , 01 April 2003 , pp. 432 - 448
Copyright
Copyright © Canadian Mathematical Society 2003

References

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