On absolutely friendly measures on

Abstract

In this paper, we extend the work of Pollington and Velani [Selecta Math. 11(2005)] to an $$-arithmetic set-up, where $$ is a finite set of valuations of $$. In particular, for an absolutely friendly measure $$ supported on a compact set in $$, we give a summation condition on an approximating function $$ such that $$ almost no point in the compact set is $$ approximable. The crucial ingredient is a version of the simplex lemma that we prove dynamically.