Weighted uniform Diophantine approximation of systems of linear forms

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet’s Theorem). An added feature is the use of general norms, rather than the supremum norm, to quantify the approximation. In terms of homogeneous dynamics, the approximation properties of an $m × n$ matrix are governed by a trajectory in $\operatorname{SL}_{m+n} (\mathbb{R}) / \operatorname{SL}_{m+n} (\mathbb{Z})$ avoiding a compact subset of the space of lattices called the critical locus defined with respect to the corresponding norm. The trajectory is formed by the action of a oneparameter diagonal subgroup corresponding to the weights. We first state a very precise form of Dirichlet’s theorem and prove it for some norms. Secondly we show, for these same norms, that the set of Dirichlet-improvable matrices has full Hausdorff dimension. Though the techniques used vary greatly depending on the chosen norm, we expect these results to hold in general.

Keywords

Dirichlet’s Theorem, Diophantine approximation with weights, geometry of numbers, dynamics on the space of lattices

2010 Mathematics Subject Classification

Primary 11J13. Secondary 11H06, 11J83, 37A17.

Full Text (PDF format)

Received 30 November 2021

Received revised 23 February 2022

Accepted 3 March 2022

Published 24 July 2022