Singular systems of linear forms and non-escape of mass in the space of lattices

Abstract

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of oneparameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdor dimension of the set of singular systems of linear forms (equivalently the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories `escaping on average' (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method di ers considerably from that of Cheung and Chevallier, and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.