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

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 Higher Moments of Arithmetic Functions in Short Intervals: A Geometric Perspective

2018 ◽  
Vol 2019 (21) ◽  
pp. 6554-6584 ◽  
Author(s):  
Daniel Rayor Hast ◽  
Vlad Matei
Keyword(s):  
Analytic Number Theory ◽  
Arithmetic Functions ◽  
Point Counting ◽  
Fixed Degree ◽  
Asymptotic Results ◽  
Counting Problem ◽  
Short Intervals ◽  
Von Mangoldt Function ◽  
Geometric Explanation ◽  
Intersection Variety

Abstract We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field $\mathbb{F}_{q}$. Using the Grothendieck–Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the ℓ-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree n in the limit as $q \to \infty $. The results of this paper can be viewed as a geometric explanation for asymptotic results that can be proved using analytic number theory over function fields.

scholarly journals Applications of the hyperbolic Ax–Schanuel conjecture

2018 ◽  
Vol 154 (9) ◽  
pp. 1843-1888 ◽  
Author(s):  
Christopher Daw ◽  
Jinbo Ren
Keyword(s):  
Abelian Varieties ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Point Counting ◽  
Counting Problem

In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the$j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax–Schanuel conjecture. In this article, we show that the hyperbolic Ax–Schanuel conjecture can be used to reduce the Zilber–Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila–Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila–Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber–Pink conjecture for curves in abelian varieties.

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