# Diophantine inheritance for p-adic measures

## TL;DR

This paper establishes $p$-adic analogues of Kleinbock's theorems on Diophantine exponents inheritance for affine subspaces, introducing a new $p$-adic exponent linked to homogeneous dynamics.

## Contribution

It proves complete $p$-adic versions of Kleinbock's theorems and confirms a conjecture of Kleinbock and Tomanov, introducing a novel $p$-adic Diophantine exponent.

## Key findings

- Confirmed a conjecture of Kleinbock and Tomanov.
- Established $p$-adic analogues of inheritance theorems.
- Introduced a new $p$-adic Diophantine exponent.

## Abstract

In this paper we prove complete $p$-adic analogues of Kleinbock's theorems \cite{Kleinbock-extremal, Kleinbock-exponent} on inheritance of Diophantine exponents for affine subspaces. In particular, we answer in the affirmative (and in a stronger form), a conjecture of Kleinbock and Tomanov \cite{KT}, as well as a question of Kleinbock \cite{Kleinbock-exponent}. Our main innovation is the introduction of a new $p$-adic Diophantine exponent which is better suited to homogeneous dynamics, and which we show to be closely related to the exponent considered by Kleinbock and Tomanov.