2016/04/14 by Kathryn Dabbs, Dabbs, Kathryn, Michael J. Kelly +3 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1604.04341
openalex publication_date 2016/04/14 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Recently Mohammadi and Salehi-Golsefidy gave necessary and sufficient\nconditions under which certain translates of homogeneous measures converge, and\nthey determined the limiting measures in the cases of convergence. The class of\nmeasures they considered includes the maximal horospherical measures. In this\npaper we prove the corresponding effective equidistribution results in the\nspace of unimodular lattices. We also prove the corresponding results for\nprobability measures with absolutely continuous densities in rank two and\nthree. Then we address the problem of determining the error terms in two\ncounting problems also considered by Mohammadi and Salehi-Golsefidy. In the\nfirst problem, we determine an error term for counting the number of lifts of a\nclosed horosphere from an irreducible, finite-volume quotient of the space of\npositive definite n\× n matrices of determinant one that intersect a ball\nwith large radius. In the second problem, we determine a logarithmic error term\nfor the Manin conjecture of a flag variety over \ℚ.\n