The exact convergence rate in the ergodic theorem of Lubotzky Phillips\n Sarnak

2018/05/14 by Antoine Pinochet-Lobos, Pinochet-Lobos, Antoine, Christophe Pittet +1
Mathematics · #37A15 and 37A30 #Advanced Algebra and Geometry #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1805.05261

openalex publication_date 2018/05/14 · openalex created_date 2022/10/01 · openalex updated_date 2026/08/04

Abstract

We compute exact convergence rates in von Neumann type ergodic theorems when\nthe acting group of measure preserving transformations is free and the means\nare taken over spheres or over balls defined by a word metric. Relying on the\nupper bounds on the spectra of Koopman operators deduced by Lubozky, Phillips,\nand Sarnak from Deligne's work on the Weil conjecture, we compute the exact\nconvergence rate for the free groups (of rank (p+1)/2 where p\≡ 1 mod 4\nis prime) of isometries of the round sphere defined by Lipschitz quaternions.\nWe also show that any finite rank free group of automorphisms of the torus\nrealizes the lowest possible discrepancy and prove a matching upper bound on\nthe convergence rate.\n

Citations

• La conjecture de Weil : I

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