On absolutely friendly measures on $\mathbb{Q}_S^d$

In this paper, we extend the work of Pollington and Velani in \cite{PV} to an $S$-arithmetic set-up, where $S$ is a finite set of valuations of $\mathbb{Q}$. In particular, for an `absolutely friendly' measure supported on a compact set in $\mathbb{Q}_S^d$, we give a summation condition on approximating function $\psi$ such that $\mu$ almost no point in the compact set is $\psi$ approximable. The crucial ingredient is a version of the simplex lemma that we prove dynamically.