Multiplicative p-Adic Approximation

Abstract

Let p be a prime number. We give several results towards a particular instance of a conjecture of Einsiedler and Kleinbock asserting that every p-adic number x satisfies

$$\underset{\mathit{a},\mathit{b}\in \mathbb{Z}\setminus \{0\}}{inf}|\mathit{a}\mathit{b}|·|\mathit{a}\mathit{x}-\mathit{b}{|}_{\mathit{p}}=0.$$

We highlight a close relationship between this conjecture and the (still open) p-adic Littlewood conjecture, according to which every real number ξ satisfies

$$\underset{\mathit{q}\in \mathbb{Z},\mathit{q}\ge 1}{inf}\mathit{q}·\Vert \mathit{q}\mathit{\xi }\Vert ·|\mathit{q}{|}_{\mathit{p}}=0.$$

Furthermore, we discuss the analogues of these conjectures over fields of power series.