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

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

Published online by Cambridge University Press:  04 June 2020

MANFRED EINSIEDLER ,
MANUEL LUETHI  and
NIMISH A. SHAH Open the ORCID record for 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
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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

equidistributionhomogeneous spaceKloosterman sumsjoining rigidityhigher-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 , Volume 41 , Issue 6 , June 2021 , pp. 1706 - 1750
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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