Classical and uniform exponents of multiplicative $p$-adic approximation

TL;DR

This paper investigates the relationships between classical and multiplicative exponents of $p$-adic approximation, establishing bounds and the range of possible ratios, and providing new inequalities for these exponents.

Contribution

It introduces bounds for the ratio of multiplicative to classical irrationality exponents and proves an upper limit for the uniform multiplicative exponent in $p$-adic numbers.

Findings

The ratio $ mu(\xi)/(\xi)$ can vary within [1, 2].

The uniform multiplicative exponent $(\xi)$ is bounded above by $(5 + )/2$.

Established bounds deepen understanding of $p$-adic approximation exponents.

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 $$ 0 < \max\{|x|, |y|\} \le X, \quad |y \xi - x|_{p} \leq X^{-\hmu} $$ has a solution in integers $x, y$ for arbitrarily large real number $X$. Its multiplicative irrationality exponent $\tmu (\xi)$ (resp., uniform multiplicative irrationality exponent $\htmu (\xi)$) is the supremum of the real numbers $\hmu$ for which the system of inequalities $$ 0 < |x y|^{1/2} \le X, \quad |y \xi - x|_{p} \leq X^{-\hmu} $$ 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 $\mu (\xi) \le \tmu(\xi) \le 2 \mu (\xi)$ and $\htmu (\xi) \le 4$. We establish that the ratio between the multiplicative irrationality exponent $\tmu$ and the irrationality exponent $\mu$ can take any given value in $[1, 2]$. Furthermore, we prove that $\htmu (\xi) \le (5 + \sqrt{5})/2$ for every $p$-adic number $\xi$.