Browsing by Author "Kadyrov S."

Now showing 1 - 3 of 3
  • Thumbnail Image
    ItemOpen Access
    Algebraic numbers, hyperbolicity, and density modulo one
    (Journal of Number Theory, 2012) Gorodnik A.; Kadyrov S.
    We prove the density of the sets of the form λm 1 μn 1ξ1 +···+ λm k μn k ξk: m,n ∈ N modulo one, where λi and μi are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelian groups.
  • Thumbnail Image
    ItemOpen Access
    ESCAPE OF MASS AND ENTROPY FOR DIAGONAL FLOWS IN REAL RANK ONE SITUATIONS
    (ISRAEL JOURNAL OF MATHEMATICS, 2014) Einsiedler M.; Kadyrov S.; Pohl A.
    Let G be a connected semisimple Lie group of real rank 1 with finite center, let Γ be a non-uniform lattice in G and a any diagonalizable element in G. We investigate the relation between the metric entropy of a acting on the homogeneous space Γ\G and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of a) which miss a fixed open set is not full.
  • Thumbnail Image
    ItemOpen Access
    SINGULAR SYSTEMS OF LINEAR FORMS AND NON-ESCAPE OF MASS IN THE SPACE OF LATTICES
    (JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 133, 2017) Kadyrov S.; Kleinbock D.; Lindenstrauss E.; Margulis G.A.
    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 one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff 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 differs considerably from that of Cheung and Chevallier and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.