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 is an element of Z\{0})vertical bar ab vertical bar.vertical bar ax-b vertical bar(p)=0. We highlight a close relationship between this conjecture and the (still open) p-adic Littlewood conjecture, according to which every real number xi satisfies (inf )(q is an element of Z,q >= 1)q.parallel to q xi parallel to.vertical bar q vertical bar(p)=0. Furthermore, we discuss the analogues of these conjectures over fields of power series.