scholarly journals Primitive Points in Rational Polygons

2020 ◽  
Vol 63 (4) ◽  
pp. 850-870
Author(s):  
Imre Bárány ◽  
Greg Martin ◽  
Eric Naslund ◽  
Sinai Robins
Keyword(s):  
Error Term ◽  
Lattice Points ◽  
Summatory Function ◽  
Primitive Lattice Points

AbstractLet ${\mathcal{A}}$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t{\mathcal{A}}$ is asymptotically $\frac{6}{\unicode[STIX]{x1D70B}^{2}}\text{Area}(t{\mathcal{A}})$ as $t\rightarrow \infty$. We show that the error term is both $\unicode[STIX]{x1D6FA}_{\pm }(t\sqrt{\log \log t})$ and $O(t(\log t)^{2/3}(\log \log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler’s $\unicode[STIX]{x1D719}(n)$.

Equidistribution of Farey sequences on horospheres in covers of and applications

2019 ◽  
Vol 41 (2) ◽  
pp. 471-493
Author(s):  
BYRON HEERSINK
Keyword(s):  
Diophantine Approximation ◽  
Lattice Points ◽  
Index Subgroup ◽  
Farey Sequence ◽  
Frobenius Numbers ◽  
Farey Sequences ◽  
Approximation Problems ◽  
Primitive Lattice Points ◽  
Finite Index Subgroup ◽  
Statistical Results

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

Two classical lattice point problems

1961 ◽  
Vol 57 (4) ◽  
pp. 699-721 ◽  
Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

scholarly journals On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

2016 ◽  
Vol 11 (2) ◽  
pp. 45-89
Author(s):  
Mordechay B. Levin
Keyword(s):  
Algebraic Number ◽  
Error Term ◽  
Main Tool ◽  
Random Variable ◽  
Algebraic Number Field ◽  
Limiting Distribution ◽  
Lattice Points ◽  
Point Problem ◽  
Lattice Point Problem ◽  
Totally Real

AbstractLet Γ ⊂ ℝs be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(θ, N) be the error term in the lattice point problem for the parallelepiped [−θ1N1, θ1N1] × . . . × [−θs Ns, θs Ns]. In this paper, we prove that ℛ(θ, N)/σ(ℛ, N) has a Gaussian limiting distribution as N→∞, where θ = (θ1, . . . , θs) is a uniformly distributed random variable in [0, 1]s, N = N1 . . . . Ns and σ(ℛ, N) ≍ (log N)(s−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the S-unit theorem.

On the Number of Primitive Lattice Points in a Parallelogram

1953 ◽  
Vol 5 ◽  
pp. 456-459 ◽  
Author(s):  
Theodor Estermann
Keyword(s):  
Real Number ◽  
Preceding Paper ◽  
Lattice Points ◽  
Primitive Lattice Points ◽  
Image Position ◽  
Positive Integers

1. Let a be any irrational real number, and let F(u) denote the number of those positive integers for which (n, [nα]) = 1. Watson proved in the preceding paper that

scholarly journals SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

2018 ◽  
Vol 6 ◽  
Author(s):  
THOMAS A. HULSE ◽  
CHAN IEONG KUAN ◽  
DAVID LOWRY-DUDA ◽  
ALEXANDER WALKER
Keyword(s):  
Error Term ◽  
Power Saving ◽  
Lattice Points ◽  
Dimensional Sphere ◽  
Mean Square ◽  
Circle Problem ◽  
Second Moments ◽  
The Laplace Transform ◽  
Error Terms ◽  
Gauss Circle Problem

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to$P_{k}(n)^{2}$, where$P_{k}(n)$is the discrepancy between the volume of the$k$-dimensional sphere of radius$\sqrt{n}$and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including$\sum P_{k}(n)^{2}e^{-n/X}$and the Laplace transform$\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions$k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums$\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral$\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.

scholarly journals Distribution of the error term for the number of lattice points inside a shifted circle

1993 ◽  
Vol 154 (3) ◽  
pp. 433-469 ◽  
Author(s):  
Pavel M. Bleher ◽  
Zheming Cheng ◽  
Freeman J. Dyson ◽  
Joel L. Lebowitz
Keyword(s):  
Error Term ◽  
Lattice Points

On two classical lattice point problems

1940 ◽  
Vol 36 (2) ◽  
pp. 131-138 ◽  
Author(s):  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

1. Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Thus P(x) is the error term in the problem of the lattice points of a circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of a rectangular hyperbola.

On differences of two squares

2006 ◽  
Vol 4 (1) ◽  
pp. 110-122
Author(s):  
Manfred Kühleitner ◽  
Werner Nowak
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Error Term ◽  
Arithmetic Function ◽  
Close Connection ◽  
Mean Square ◽  
Number Of Divisors ◽  
Summatory Function ◽  
Average Size ◽  
Sharp Lower Bound

AbstractThe arithmetic function ρ(n) counts the number of ways to write a positive integer n as a difference of two squares. Its average size is described by the Dirichlet summatory function Σn≤x ρ(n), and in particular by the error term R(x) in the corresponding asymptotics. This article provides a sharp lower bound as well as two mean-square results for R(x), which illustrates the close connection between ρ(n) and the number-of-divisors function d(n).

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