2024 CLASSICAL AND UNIFORM EXPONENTS OF MULTIPLICATIVE $p$-ADIC APPROXIMATION
Yann Bugeaud, Johannes Schleischitz
Publ. Mat. 68(1): 3-26 (2024). DOI: 10.5565/PUBLMAT6812401

## Abstract

Let $p$ be a prime number and $\xi$ an irrational $p$-adic number. Its irrationality exponent $\mu (\xi)$ is the supremum of the real numbers $\mu$ for which the system of inequalities

$0<\max\{\vert x\vert,\vert y\vert\}\leq X,\quad\vert y\xi-x\vert_p\leq X^{-\mu}$

has a solution in integers $x$, $y$ for arbitrarily large real number $X$. Its multiplicative irrationality exponent $\mu^{\times} (\xi)$ (resp., uniform multiplicative irrationality exponent $\widehat{\mu}^{\times} (\xi)$ is the supremum of the real numbers $\widehat\mu$ for which the system of inequalities

$0\<\vert xy\vert^{1/2}\leq X,\quad\vert y\xi-x\vert_p\leq X^{-\widehat\mu}$

has a solution in integers $x$, $y$ for arbitrarily large (resp., for every sufficiently large) real number $X$. It is not difficult to show that $\mu (\xi) \le \mu^{\times}(\xi) \le 2 \mu (\xi)$ and $\widehat{\mu}^{\times} (\xi) \le 4$. We establish that the ratio between the multiplicative irrationality exponent $\mu^{\times}$ and the irrationality exponent $\mu$ can take any given value in $\lbrack1,\;2\rbrack$. Furthermore, we prove that $\widehat{\mu}^{\times} (\xi) \le (5 + \sqrt{5})/2$ for every $p$-adic number $\xi$.

## Acknowledgements

The authors are very grateful to the referees,whose numerous detailed remarks and corrections helped them to considerably improve the presentation of the paper.

## Citation

Yann Bugeaud. Johannes Schleischitz. "CLASSICAL AND UNIFORM EXPONENTS OF MULTIPLICATIVE $p$-ADIC APPROXIMATION." Publ. Mat. 68 (1) 3 - 26, 2024. https://doi.org/10.5565/PUBLMAT6812401

## Information

Received: 26 May 2021; Accepted: 1 October 2021; Published: 2024
First available in Project Euclid: 25 December 2023

MathSciNet: MR4682721
Digital Object Identifier: 10.5565/PUBLMAT6812401

Subjects:
Primary: 11J04 , 11J61

Keywords: exponent of approximation , p-adic number , Rational approximation