Diophantine approximation on manifolds and lower bounds for Hausdorff dimension

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.