Abstract
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We establish effective counting results for lattice points in families of domains in real,
complex and quaternionic hyperbolic spaces of any dimension. The domains we focus
on are defined as product sets with respect to an Iwasawa decomposition. Several
natural diophantine problems can be reduced to counting lattice points in such domains.
These include equidistribution of the ratio of the length of the shortest solution
to the gcd
equation
relative
to the length of
,
where
ranges over primitive vectors in a disc whose radius increases, the natural analog of
this problem in imaginary quadratic number fields, as well as equidistribution of
integral solutions to the diophantine equation defined by an integral Lorentz form
in three or more variables. We establish an effective rate of convergence
for these equidistribution problems, depending on the size of the spectral
gap associated with a suitable lattice subgroup in the isometry group of
the relevant hyperbolic space. The main result underlying our discussion
amounts to establishing effective joint equidistribution for the horospherical
component and the radial component in the Iwasawa decomposition of lattice
elements.
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Keywords
hyperbolic spaces, horospherical coordinates,
equidistribution of lattice points, spectral gap
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Mathematical Subject Classification
Primary: 22E30, 22E40
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Milestones
Received: 24 October 2021
Revised: 29 December 2022
Accepted: 29 January 2023
Published: 26 July 2023
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| © 2023 The Author(s), under
exclusive license to MSP (Mathematical Sciences Publishers).
Distributed under the Creative Commons
Attribution License 4.0 (CC BY). |
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