Diophantine approximation on manifolds and lower bounds for Hausdorff dimension

Abstract

Given $n\in \mathbb{N}$ and $\tau >\frac{1}{n}$, let ${\mathcal{S}}_n(\tau )$ denote the classical set of $\tau$-approximable points in ${\mathbb{R}}^n$, which consists of $\mathbf{x}\in {\mathbb{R}}^n$ that lie within distance $q^{-\tau -1}$ from the lattice $\frac{1}{q}{\mathbb{Z}}^n$ 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 $\tau >\frac{1}{n}$ 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 $\frac{1}{n}\le \tau <\frac{1}{m}$. 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.

20 pages

References in corpus (1)

• Rational points near planar curves and Diophantine approximation