Best possible rates of distribution of dense lattice orbits in homogeneous spaces

Anish Ghosh, Alexander Gorodnik, Amos Nevo

Research output: Contribution to journalArticle (Academic Journal)peer-review

5 Citations (Scopus)
224 Downloads (Pure)

Abstract

This paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup ⌈ in a connected Lie (or algebraic) group G, acting on suitable homogeneous spaces G/H. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on H and acting on G/⌈. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of H in the automorphic representation on L2(⌈\G). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.

Original languageEnglish
Pages (from-to)155-188
Number of pages34
JournalJournal fur die Reine und Angewandte Mathematik
Volume2018
Issue number745
DOIs
Publication statusPublished - 10 Jul 2014

Keywords

  • Diophantine approximation
  • semisimple algebraic group
  • homogeneous space
  • lattice subgroup
  • automorphic spectrum

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