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Ergodic Theory and Dynamical Systems Ergodic Theory and Dynamical Systems

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Primitive rational points on expanding horocycles in products of the modular surface with the torus

Part of: Ergodic theory Exponential sums and character sums Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  04 June 2020

MANFRED EINSIEDLER ,
MANUEL LUETHI  and
NIMISH A. SHAH [Opens in a new window]
MANFRED EINSIEDLER
Affiliation:
D-Math, ETH Zürich, Rämistrasse 101, CH-8092Zürich, Switzerland email manfred.einsiedler@math.ethz.ch
MANUEL LUETHI
Affiliation:
Department of Mathematics, University of Tel Aviv, 69978Tel-Aviv, Israel email manuelluthi@mail.tau.ac.il
NIMISH A. SHAH
Affiliation:
The Ohio State University, Columbus, OH 43210, USA email shah@math.ohio-state.edu
Corresponding
E-mail address:
manfred.einsiedler@math.ethz.ch
manuelluthi@mail.tau.ac.il
shah@math.ohio-state.edu
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Abstract

We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface. This answers a question posed in joint work by the first and the last named author with Shahar Mozes and Uri Shapira. Under certain congruence conditions we prove the joint equidistribution of conjugate rational points in the $2$-torus and the modular surface.

Keywords

equidistribution homogeneous space Kloosterman sums joining rigidity higher-rank action

MSC classification

Primary: 37A45: Relations with number theory and harmonic analysis 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 11L05: Gauss and Kloosterman sums; generalizations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 45
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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