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$ \Omega$-results for the hyperbolic lattice point problem


Author: Dimitrios Chatzakos
Journal: Proc. Amer. Math. Soc. 145 (2017), 1421-1437
MSC (2010): Primary 11F72; Secondary 37C35, 37D40
DOI: https://doi.org/10.1090/proc/13348
Published electronically: October 24, 2016
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Abstract: For $ \Gamma $ a cocompact or cofinite Fuchsian group, we study the lattice point problem on the Riemann surface $ \Gamma \backslash \mathbb{H}$. The main asymptotic for the counting of the orbit $ \Gamma z$ inside a circle of radius $ r$ centered at $ z$ grows like $ c e^r$. Phillips and Rudnick studied $ \Omega $-results for the error term and mean results in $ r$ for the normalized error term. We investigate the normalized error term in the natural parameter $ X=2 \cosh r$ and prove $ \Omega _{\pm }$-results for the orbit $ \Gamma w$ and circle centered at $ z$, even for $ z \neq w$.


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Additional Information

Dimitrios Chatzakos
Affiliation: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
Email: d.chatzakos.12@ucl.ac.uk

DOI: https://doi.org/10.1090/proc/13348
Received by editor(s): December 14, 2015
Received by editor(s) in revised form: May 11, 2016, and May 24, 2016
Published electronically: October 24, 2016
Additional Notes: The author was supported by a DTA from EPSRC during his Ph.D. studies at UCL
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2016 American Mathematical Society