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

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 x n matrices are both equal to m(1-1/m+n), thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis as well as answering a question of Bugeaud, Cheung, and Chevallier. Other applications include computing the dimensions of the sets of points witnessing conjectures of Starkov and Schmidt. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.