variance of lattice point counting in some special shells in Rd

2021 ◽  
Author(s):  
Tao Jiang
Keyword(s):  
Lattice Point ◽  
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.

scholarly journals The lattice point counting problem on the Heisenberg groups

2015 ◽  
Vol 65 (5) ◽  
pp. 2199-2233 ◽  
Author(s):  
Rahul Garg ◽  
Amos Nevo ◽  
Krystal Taylor
Keyword(s):  
Lattice Point ◽  
Heisenberg Groups ◽  
Point Counting ◽  
Counting Problem

Polynomial values of surface point counting polynomials

2020 ◽  
pp. 1-18
Author(s):  
L. Hajdu ◽  
O. Herendi
Keyword(s):  
Lattice Point ◽  
Lattice Points ◽  
Surface Point ◽  
Point Counting ◽  
Dimensional Cube

There are many results in the literature concerning power values, equal values or more generally, polynomial values of lattice point counting polynomials. In this paper, we prove various finiteness results for polynomial values of polynomials counting the lattice points on the surface of an [Formula: see text]-dimensional cube, pyramid and simplex.

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