Good functions, measures, and the Kleinbock–Tomanov conjecture

Abstract

Abstract In this paper, we prove a conjecture of Kleinbock and Tomanov (2007) on Diophantine properties of a large class of fractal measures on Q p n \mathbb{Q}_{p}^{n} . More generally, we establish the 𝑝-adic analogues of the influential results of Kleinbock, Lindenstrauss and Weiss (2004) on Diophantine properties of friendly measures. We further prove the 𝑝-adic analogue of one of the main results of Kleinbock (2008) concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of Kleinbock, Lindenstrauss and Weiss is a result on ( C , α ) (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 , α ) (C,\alpha) -good in the 𝑝-adic setting. We believe this result will be of independent interest.