Badly approximable vectors on rational quadratic varieties

TL;DR

This paper studies badly approximable vectors on rational quadratic varieties, proving they form large, winning sets with full Hausdorff dimension, contrasting their typical measure-zero nature.

Contribution

It introduces the concept of badly approximable vectors on rational quadratic varieties and proves these sets are winning and have full Hausdorff dimension.

Findings

Sets are winning and have full Hausdorff dimension.

Their intersections also have full Hausdorff dimension.

These sets are null sets in measure, highlighting their fractal complexity.

Abstract

Approximation in this paper is of vectors on the unit $d$-cube by the projection of integer lattice points onto the same cube. We define badly approximable vectors on a rational quadratic variety and show that sets of these vectors, which are (naturally) indexed by $m \in \QQ$, are winning and strong winning in the sense of Schmidt games. From the winning property, it follows that these sets have full Hausdorff dimension and, moreover, so does their intersection. In most cases, these sets are known to be null sets.