On the packing dimension of weighted singular matrices on fractals

TL;DR

This paper establishes new upper bounds for the packing dimension of weighted singular matrices on fractals, expanding known results to broader classes of fractals and matrices.

Contribution

It introduces the first known upper bounds for the packing dimension of weighted singular matrices and extends these bounds to intersections with fractals.

Findings

Upper bounds for packing dimension of weighted singular matrices.

Bounds for intersections with fractal subsets.

Extension of results to broader classes of fractals and vectors.

Abstract

We provide the first known upper bounds for the packing dimension of weighted singular and weighted $\omega$-singular matrices. We also prove upper bounds for these sets when intersected with fractal subsets. The latter results, even in the unweighted setting, are already new for matrices. Further, even for row vectors, our results enlarge the class of fractals for which bounds are currently known. We use methods from homogeneous dynamics, in particular we provide upper bounds for the packing dimension of points on the space of unimodular lattices, whose orbits under diagonal flows $p$-escape on average.