Simultaneous visibility in the integer lattice

Daniel Berend; Rishi Kumar; Andrew Pollington

doi:10.1016/j.jnt.2023.03.003

Simultaneous visibility in the integer lattice

Daniel Berend, Rishi Kumar, Andrew Pollington

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Two lattice points are visible from one another if there is no lattice point on the open line segment joining them. Let S be a finite subset of Z^k. The asymptotic density of the set of lattice points, visible from all points of S, was studied by several authors. Our main result is an improved upper bound on the error term. We also find the Schnirelmann density of the set of visible points from some sets S. Finally, we discuss these questions from the point of view of ergodic theory.

Original language	English
Pages (from-to)	84-123
Number of pages	40
Journal	Journal of Number Theory
Volume	250
DOIs	https://doi.org/10.1016/j.jnt.2023.03.003
State	Published - 1 Sep 2023

Keywords

Schnirelmann densities
Selberg's sieve
Simultaneous visibility
Unique ergodicity

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jnt.2023.03.003

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title = "Simultaneous visibility in the integer lattice",

abstract = "Two lattice points are visible from one another if there is no lattice point on the open line segment joining them. Let S be a finite subset of Zk. The asymptotic density of the set of lattice points, visible from all points of S, was studied by several authors. Our main result is an improved upper bound on the error term. We also find the Schnirelmann density of the set of visible points from some sets S. Finally, we discuss these questions from the point of view of ergodic theory.",

keywords = "Schnirelmann densities, Selberg's sieve, Simultaneous visibility, Unique ergodicity",

author = "Daniel Berend and Rishi Kumar and Andrew Pollington",

note = "Funding Information: Research supported in part by the Milken Families Foundation Chair in Mathematics. Publisher Copyright: {\textcopyright} 2023 Elsevier Inc.",

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AU - Kumar, Rishi

AU - Pollington, Andrew

PY - 2023/9/1

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N2 - Two lattice points are visible from one another if there is no lattice point on the open line segment joining them. Let S be a finite subset of Zk. The asymptotic density of the set of lattice points, visible from all points of S, was studied by several authors. Our main result is an improved upper bound on the error term. We also find the Schnirelmann density of the set of visible points from some sets S. Finally, we discuss these questions from the point of view of ergodic theory.

AB - Two lattice points are visible from one another if there is no lattice point on the open line segment joining them. Let S be a finite subset of Zk. The asymptotic density of the set of lattice points, visible from all points of S, was studied by several authors. Our main result is an improved upper bound on the error term. We also find the Schnirelmann density of the set of visible points from some sets S. Finally, we discuss these questions from the point of view of ergodic theory.

KW - Schnirelmann densities

KW - Selberg's sieve

KW - Simultaneous visibility

KW - Unique ergodicity

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VL - 250

SP - 84

EP - 123

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

Simultaneous visibility in the integer lattice

Abstract

Keywords

ASJC Scopus subject areas

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