scholarly journals Primitive lattice points in a thin strip along the boundary of a large convex planar domain

2001 ◽  
Vol 99 (4) ◽  
pp. 331-341 ◽  
Author(s):  
Ekkehard Krätzel ◽  
Werner Georg Nowak
Keyword(s):  
Lattice Points ◽  
Planar Domain ◽  
Thin Strip ◽  
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

An Ω-estimate for the lattice rest of a convex planar domain

1985 ◽  
Vol 100 (3-4) ◽  
pp. 295-299 ◽  
Author(s):  
Werner Georg Nowak
Keyword(s):  
Classical Result ◽  
Compact Convex ◽  
Lattice Points ◽  
Planar Domain ◽  
Circle Problem ◽  
Asymptotic Formulae ◽  
Lattice Rest ◽  
Image Position

SynopsisLet D be a compact convex planar domain containing the origin, the boundary of which is of class C∞ and has finite non-vanishing curvature throughout. For the number A(i) of lattice points in the “blown up” domain √tD, the estimateis established. This is a generalization of Hardy's classical result for the circle problem. The proof is based on asymptotic formulae for certain exponential integrals due to E. Hlawka.

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
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