Good functions, measures, and the Kleinbock-Tomanov conjecture

TL;DR

This paper proves a conjecture on Diophantine properties of fractal measures in the $p$-adic setting, extending key results of Kleinbock, Lindenstrauss, and Weiss, and introduces a new method for establishing $(C, abla)$-good functions.

Contribution

It establishes the $p$-adic analogues of major Diophantine results for fractal and friendly measures, and introduces a novel approach to proving functions are $(C, abla)$-good in the $p$-adic context.

Findings

Proved Kleinbock-Tomanov conjecture for $p$-adic fractal measures.

Extended Diophantine results to $p$-adic friendly measures.

Developed a new method for $(C, abla)$-good functions in $p$-adic analysis.

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.