Entropy in the cusp and singular systems of linear forms

Abstract

Abstract. Singular systems of linear forms were introduced by Khint-chine 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 one-parameter diagonal groups on the space of lattices. We give a (con-jecturally sharp) upper bound on the Hausdorff dimension of singular systems of linear forms (equivalently the set of points with divergent trajectories) as well as the dimension of the set of points with trajec-tories ‘escaping on average ’ (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung, on the vector case. Our method differs considerably from that of Cheung and Chevallier, and is based on the method of integral inequalities developed by Eskin, Margulis and Mozes. 1.

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