Good functions, measures, and the Kleinbock-Tomanov conjecture

Abstract

In this paper we prove a conjecture of Kleinbock and Tomanov \cite[Conjecture~FP]{KT} on Diophantine properties of a large class of fractal measures on $\mathbb{Q}_p^n$. More generally, we establish the $p$-adic analogues of the influential results of Kleinbock, Lindenstrauss, and Weiss \cite{KLW} on Diophantine properties of friendly measures. We further prove the $p$-adic analogue of one of the main results in \cite{Kleinbock-exponent} due to Kleinbock concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of \cite{KLW} is a result on $(C, \alpha)$-good functions whose proof crucially uses the mean value theorem. Our main technical innovation is an alternative approach to establishing that certain functions are $(C, \alpha)$-good in the $p$-adic setting. We believe this result will be of independent interest.