Diophantine approximation on manifolds and lower bounds for Hausdorff dimension

TL;DR

This paper establishes sharp lower bounds for the Hausdorff dimension of sets of well-approximable points on smooth manifolds, using a new approach based on the Mass Transference Principle, extending previous results in Diophantine approximation.

Contribution

It introduces a novel method leveraging the Mass Transference Principle to determine precise lower bounds for Hausdorff dimension on manifolds, improving understanding of Diophantine approximation on submanifolds.

Findings

Provides sharp lower bounds for Hausdorff dimension of approximation sets

Shows the bounds are optimal for the given conditions

Extends results to general approximating functions

Abstract

Given $n\in\mathbb{N}$ and $\tau>\frac1n$, let $\mathcal{S}_n(\tau)$ denote the classical set of $\tau$-approximable points in $\mathbb{R}^n$, which consists of ${\bf x}\in \mathbb{R}^n$ that lie within distance $q^{-\tau-1}$ from the lattice $\frac1q\mathbb{Z}^n$ for infinitely many $q\in\mathbb{N}$. In pioneering work, Kleinbock $\&$ Margulis showed that for any non-degenerate submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $\tau>\frac1n$ almost all points on $\mathcal{M}$ are not $\tau$-approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set $\mathcal{M}\cap\mathcal{S}_n(\tau)$. In this paper we suggest a new approach based on the Mass Transference Principle, which enables us to find a sharp lower bound for $\dim \mathcal{M}\cap\mathcal{S}_n(\tau)$ for any $C^2$ submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $\tau$ satisfying $\frac1n\le\tau<\frac1m$. Here $m$ is the codimension of $\mathcal{M}$. We also show that the condition on $\tau$ is best possible and extend the result to general approximating functions.