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 infa,b∈Z∖{0}|ab|·|ax−b|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 infq∈Z,q≥1q·‖qξ‖·|q|p=0. Furthermore, we discuss the analogues of these conjectures over fields of power series.