Badly approximable points on manifolds and unipotent orbits in homogeneous spaces

TL;DR

This paper proves that the intersection of weighted badly approximable points on certain manifolds has full Hausdorff dimension, using homogeneous dynamics and unipotent orbit distribution techniques.

Contribution

It extends previous results by removing weight restrictions and smoothness conditions, employing homogeneous dynamics to analyze badly approximable points on manifolds.

Findings

Full Hausdorff dimension of intersections of weighted badly approximable points

Removal of weight restrictions in approximation results

Application of unipotent orbit distribution in homogeneous spaces

Abstract

In this paper, we study the weighted $n$-dimensional badly approximable points on manifolds. Given a $C^n$ differentiable non-degenerate submanifold $\mathcal{U} \subset \mathbb{R}^n$, we will show that any countable intersection of the sets of the weighted badly approximable points on $\mathcal{U}$ has full Hausdorff dimension. This strengthens a result of Beresnevich by removing the condition on weights and weakening the smoothness condition on manifolds. Compared to the work of Beresnevich, our approach relies on homogeneous dynamics. It turns out that in order to solve this problem, it is crucial to study the distribution of long pieces of unipotent orbits in homogeneous spaces. The proof relies on the linearization technique and representations of $\mathrm{SL}(n+1,\mathbb{R})$.