Primitive lattice points in rational ellipses and related arithmetic functions

1988 ◽  
Vol 106 (1) ◽  
pp. 57-63 ◽  
Author(s):  
Werner Georg Nowak
Keyword(s):  
Lattice Points ◽  
Arithmetic Functions ◽  
Primitive Lattice Points

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 $\mathbf{A}$.

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 Primitive lattice points in convex planar domains

1996 ◽  
Vol 76 (3) ◽  
pp. 271-283 ◽  
Author(s):  
Martin Huxley ◽  
Werner Nowak
Keyword(s):  
Lattice Points ◽  
Planar Domains ◽  
Primitive Lattice Points

scholarly journals Primitive lattice points in planar domains

2010 ◽  
Vol 142 (3) ◽  
pp. 267-302 ◽  
Author(s):  
Roger C. Baker
Keyword(s):  
Lattice Points ◽  
Planar Domains ◽  
Primitive Lattice Points

scholarly journals On primitive lattice points in planar domains

2003 ◽  
Vol 109 (1) ◽  
pp. 1-26 ◽  
Author(s):  
Wenguang Zhai
Keyword(s):  
Lattice Points ◽  
Planar Domains ◽  
Primitive Lattice Points

On the Distribution of Primitive Lattice Points in the Plane

1959 ◽  
Vol 2 (2) ◽  
pp. 91-96 ◽  
Author(s):  
J.H.H. Chalk ◽  
P. Erdos
Keyword(s):  
Lattice Points ◽  
Real Numbers ◽  
Rational Field ◽  
Linearly Independent ◽  
Primitive Lattice Points ◽  
Image Position

Let 1,θ1, θ2, …,θn be real numbers linearly independent over the rational field and let α1, α2,…, αn be arbitrary real numbers. Then, to each N > 0 and ε > 0, there correspond integerswhich satisfy the set of inequalitiesA

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 $\unicode[STIX]{x1D719}(n)$.

Sign in / Sign up

Export Citation Format

Share Document