# A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

## TL;DR

This paper introduces a variational principle linking metric Diophantine approximation with the parametric geometry of numbers, enabling the calculation of dimensions of sets related to matrix approximation and resolving several conjectures.

## Contribution

It establishes a new variational principle and introduces the concept of templates, generalizing rigid systems, to compute dimensions of sets in Diophantine approximation.

## Key findings

- Proved the Hausdorff and packing dimensions of singular matrices are both mn(1-1/(m+n)).
- Resolved a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis (2014).
- Provided a framework to compute dimensions of sets related to Starkov and Schmidt conjectures.

## Abstract

We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular $m\times n$ matrices are both equal to $mn \big(1-\frac1{m+n}\big)$, thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis (preprint 2014) as well as answering a question of Bugeaud, Cheung, and Chevallier (preprint 2016). We introduce the notion of a $template$, which generalizes the notion of a $rigid$ $system$ (Roy, 2015) to the setting of matrix approximation. Our main theorem takes the following form: for any class of templates $\mathcal F$ closed under finite perturbations, the Hausdorff and packing dimensions of the set of matrices whose successive minima functions are members of $\mathcal F$ (up to finite perturbation) can be written as the suprema over $\mathcal F$ of certain natural functions on the space of templates. Besides implying KKLM's conjecture, this theorem has many other applications including computing the Hausdorff and packing dimensions of the set of points witnessing a conjecture of Starkov (2000), and of the set of points witnessing a conjecture of Schmidt (1983).