$\Omega$-results for the hyperbolic lattice point problem
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- by Dimitrios Chatzakos PDF
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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 $z \neq w$.References
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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
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1421-1437
- MSC (2010): Primary 11F72; Secondary 37C35, 37D40
- DOI: https://doi.org/10.1090/proc/13348
- MathSciNet review: 3601536