Fractional dimension related to badly approximable matrices associated with higher successive minima

TL;DR

This paper introduces a new concept of badly approximable matrices of higher order using successive minima, showing they have measure zero but full Hausdorff dimension for certain orders.

Contribution

It extends the theory of badly approximable matrices by defining higher order versions and analyzing their measure and dimension properties.

Findings

Badly approximable matrices of order less than d have Lebesgue measure zero.

Gaps between these matrices still possess full Hausdorff dimension.

Abstract

In this article we introduce the notion of badly approximable matrices of higher order using higher sucessive minima in $\mathbb R^d$. We prove that for order less than $d$, they have Lebesgue measure zero and the gaps between them still have full Hausdorff dimension.