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

Abstract

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