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

Ghosh, Anish ; Gorodnik, Alexander ; Nevo, Amos (2016) Best possible rates of distribution of dense lattice orbits in homogeneous spaces Journal fur die reine und angewandte Mathematik . No pp. given. ISSN 0075-4102

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Official URL: https://www.degruyter.com/view/j/crll.ahead-of-pri...

Related URL: http://dx.doi.org/10.1515/crelle-2016-0001

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 Γ 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/Γ. 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(Γ ∖ 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.

Item Type:Article
Source:Copyright of this article belongs to Walter de Gruyter GmbH & Co. KG.
ID Code:114621
Deposited On:05 Jun 2018 06:56
Last Modified:05 Jun 2018 06:56

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