scholarly journals Hyperbolic lattice-point counting and modular symbols

2009 ◽  
Vol 21 (3) ◽  
pp. 721-734 ◽  
Author(s):  
Yiannis N. Petridis ◽  
Morten S. Risager
Keyword(s):  
Lattice Point ◽  
Point Counting ◽  
Modular Symbols ◽  
Hyperbolic Lattice ◽  
Hyperbolic Lattice Point Counting

On Representation of Integers from Thin Subgroups of SL(2, ℤ) with Parabolics

2018 ◽  
Vol 2020 (18) ◽  
pp. 5611-5629 ◽  
Author(s):  
Xin Zhang
Keyword(s):  
Critical Exponent ◽  
Lattice Point ◽  
Fuchsian Group ◽  
Spectral Gap ◽  
Circle Method ◽  
Inner Product ◽  
Point Counting ◽  
Finitely Generated ◽  
Exceptional Set ◽  
Representation Of Integers

Abstract Let $\Lambda <SL(2,\mathbb{Z})$ be a finitely generated, nonelementary Fuchsian group of the 2nd kind, and $\mathbf{v},\mathbf{w}$ be two primitive vectors in $\mathbb{Z}^2\!-\!\mathbf{0}$. We consider the set $\mathcal{S}\!=\!\{\left \langle \mathbf{v}\gamma ,\mathbf{w}\right \rangle _{\mathbb{R}^2}\!:\!\gamma\! \in\! \Lambda \}$, where $\left \langle \cdot ,\cdot \right \rangle _{\mathbb{R}^2}$ is the standard inner product in $\mathbb{R}^2$. Using Hardy–Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich, and Sarnak, together with Gamburd’s 5/6 spectral gap, we show that if $\Lambda $ has parabolic elements, and the critical exponent $\delta $ of $\Lambda $ exceeds 0.998317, then a density-one subset of all admissible integers (i.e., integers passing all local obstructions) are actually in $\mathcal{S}$, with a power savings on the size of the exceptional set (i.e., the set of admissible integers failing to appear in $\mathcal{S}$). This supplements a result of Bourgain–Kontorovich, which proves a density-one statement for the case when $\Lambda $ is free, finitely generated, has no parabolics, and has critical exponent $\delta>0.999950$.

scholarly journals Effective lattice point counting in rational convex polytopes

2004 ◽  
Vol 38 (4) ◽  
pp. 1273-1302 ◽  
Author(s):  
Jesús A. De Loera ◽  
Raymond Hemmecke ◽  
Jeremiah Tauzer ◽  
Ruriko Yoshida
Keyword(s):  
Lattice Point ◽  
Convex Polytopes ◽  
Point Counting

scholarly journals Uniform distribution and lattice point counting

1992 ◽  
Vol 53 (1) ◽  
pp. 39-50 ◽  
Author(s):  
G. R. Everest
Keyword(s):  
Asymptotic Formula ◽  
Uniform Distribution ◽  
Lattice Point ◽  
Lattice Points ◽  
Analytic Method ◽  
Point Counting

AbstractA well-known theorem of Hardy and Littlewood gives a three-term asymptotic formula, counting the lattice points inside an expanding, right triangle. In this paper a generalisation of their theorem is presented. Also an analytic method is developed which enables one to interpret the coefficients in the formula. These methods are combined to give a generalisation of a “heightcounting” formula of Györy and Pethö which itself was a generalisation of a theorem of Lang.

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