On simultaneous uniform approximation to a 𝑝-adic number and its square

Bugeaud, Yann

doi:10.1090/S0002-9939-2010-10491-4

On simultaneous uniform approximation to a $p$-adic number and its square
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by Yann Bugeaud

Proc. Amer. Math. Soc. 138 (2010), 3821-3826

DOI: https://doi.org/10.1090/S0002-9939-2010-10491-4

Published electronically: May 21, 2010

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Abstract:

Let $p$ be a prime number. We show that a result of Teulié is nearly best possible by constructing a $p$-adic number $\xi$ such that $\xi$ and $\xi ^2$ are uniformly simultaneously very well approximable by rational numbers with the same denominator. The same conclusion was previously reached by Zelo in his PhD thesis, but our approach using $p$-adic continued fractions is more direct and simpler.

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