Multiplicative p -Adic Approximation

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 inf a,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 inf q∈Z,q⩾1q⋅‖qξ‖⋅|q|p=0. Furthermore, we discuss the analogues of these conjectures over fields of power series.