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.