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Good functions, measures, and the Kleinbock-Tomanov conjecture

Victor Beresnevich, Shreyasi Datta, Anish Ghosh

TL;DR

This work settles Kleinbock–Tomanov's conjecture FP in the $p$-adic context by extending the Diophantine theory of friendly measures to $\mathbb{Q}_p^n$. The authors develop a novel $p$-adic approach to $(C,\alpha)$-goodness that avoids the Mean Value Theorem, using higher-order difference quotients and carefully controlled polynomial approximations. They establish that pushforwards of absolutely decaying Federer measures under suitably nondegenerate $C^{l+1}$ maps are friendly, deducing strong extremality for self-similar measures and the $p$-adic analogue of affine-subspace Diophantine inheritance. Furthermore, they prove the $p$-adic version of Kleinbock–exponent results, showing exact Diophantine exponents for pushforwards onto affine subspaces. Collectively, these results extend the KLW framework to $p$-adic and $S$-arithmetic settings, providing new tools and potential applications to Diophantine approximation over ultrametric fields and related domains.

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 $(C, α)$-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.

Good functions, measures, and the Kleinbock-Tomanov conjecture

TL;DR

This work settles Kleinbock–Tomanov's conjecture FP in the -adic context by extending the Diophantine theory of friendly measures to . The authors develop a novel -adic approach to -goodness that avoids the Mean Value Theorem, using higher-order difference quotients and carefully controlled polynomial approximations. They establish that pushforwards of absolutely decaying Federer measures under suitably nondegenerate maps are friendly, deducing strong extremality for self-similar measures and the -adic analogue of affine-subspace Diophantine inheritance. Furthermore, they prove the -adic version of Kleinbock–exponent results, showing exact Diophantine exponents for pushforwards onto affine subspaces. Collectively, these results extend the KLW framework to -adic and -arithmetic settings, providing new tools and potential applications to Diophantine approximation over ultrametric fields and related domains.

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 . More generally, we establish the -adic analogues of the influential results of Kleinbock, Lindenstrauss, and Weiss \cite{KLW} on Diophantine properties of friendly measures. We further prove the -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 -good functions whose proof crucially uses the mean value theorem. Our main technical innovation is an alternative approach to establishing that certain functions are -good in the -adic setting. We believe this result will be of independent interest.
Paper Structure (28 sections, 27 theorems, 189 equations)

This paper contains 28 sections, 27 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Difficulties in $p$-adic fields
  4. Preliminaries
  5. Besicovitch spaces
  6. Federer measures
  7. Nonplanar measures
  8. $C^k$ functions in $\mathbb{Q}_\nu^d$
  9. Nondegeneracy in an affine subspace
  10. Good maps and decaying measures
  11. Friendly measures
  12. Diophantine exponents
  13. Proofs of \ref{['ff']}, \ref{['FP']} and \ref{['DG']}
  14. Proof of \ref{['ff']} using \ref{['mainp']} and \ref{['lemmmm']}
  15. Proof of \ref{['FP']} using \ref{['ff']}
  16. ...and 13 more sections

Key Result

Theorem 2.1

Let $U$ be an open subset of $\mathbb{Q}_\nu^d$, and $\mathbf{f}: U\to \mathbb{Q}_\nu^n$. Let $\mu$ be an absolutely decaying Federer measure on $U$. Suppose that $\mathbf{f}$ is a $C^{l+1}$ map that is nonsingular and $l$-nondegenerate at $\mu$-almost every point. Then $\mathbf{f}_\star\mu$ is frie

Theorems & Definitions (49)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Definition 3.1
  • Remark 3.1
  • Proposition 3.1: See KM and KT
  • Corollary 3.1
  • Definition 3.2: Friendly measure
  • Proposition 3.2
  • ...and 39 more