Good functions, measures, and the Kleinbock-Tomanov conjecture

In this paper we prove a conjecture of Kleinbock and Tomanov on Diophantine properties of a large class of fractal measures on Qnp. More generally, we establish the p-adic analogues of the influential results of Kleinbock, Lindenstrauss, and Weiss on Diophantine properties of friendly measures. We further prove the p-adic analogue of Kleinbock's theorem concerning Diophantine inheritance of affine subspaces. One of the key ingredients in the proofs of Kleinbock, Lindenstrauss, and Weiss is a result on (C,α)-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)-good in the p-adic setting. We believe this result will be of independent interest.