Generalized lattice-point visibility in ℕk

Benedetti, Carolina; Estupiñan, Santiago; Harris, Pamela E.

doi:10.2140/involve.2021.14.103

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Generalized lattice-point visibility in $\mathbb{N}^k$

Carolina Benedetti, Santiago Estupiñan and Pamela E. Harris

Vol. 14 (2021), No. 1, 103–118

DOI: 10.2140/involve.2021.14.103

Abstract

A lattice point $(r, s) \in ℕ^{2}$ is said to be visible from the origin if no other integer lattice point lies on the line segment joining the origin and $(r, s)$ . It is a well-known result that the proportion of lattice points visible from the origin is given by $1 ∕ ζ (2)$ , where $ζ (s) = \sum_{n = 1}^{\infty} 1 ∕ n^{s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika generalized the notion of lattice-point visibility by saying that for a fixed $b \in ℕ$ a lattice point $(r, s) \in ℕ^{2}$ is $b$ -visible from the origin if no other lattice point lies on the graph of a function $f (x) = m x^{b}$ , for some $m \in ℚ$ , between the origin and $(r, s)$ . In their analysis they establish that for a fixed $b \in ℕ$ the proportion of $b$ -visible lattice points is $1 ∕ ζ (b + 1)$ , which generalizes the result in the classical lattice-point visibility setting. In this paper we give an $n$ -dimensional notion of $b$ -visibility that recovers the one presented by Goins et. al. in two dimensions, and the classical notion in $n$ dimensions. We prove that for a fixed $b = (b_{1}, b_{2}, \dots, b_{n}) \in ℕ^{n}$ the proportion of $b$ -visible lattice points is given by $1 ∕ ζ (\sum_{i = 1}^{n} b_{i})$ .

Moreover, we give a new notion of $b$ -visibility for vectors

b = (b_{1} ∕ a_{1}, b_{2} ∕ a_{2}, \dots, b_{n} ∕ a_{n}) \in {(ℚ ∖ {0})}^{n},

with nonzero rational entries. In this case, our main result establishes that the proportion of $b$ -visible points is $1 ∕ ζ (\sum_{i \in J} | b_{i} |)$ , where $J$ is the set of the indices $1 \leq i \leq n$ for which $b_{i} ∕ a_{i} < 0$ . This result recovers a main theorem of Harris and Omar for $b \in ℚ ∖ {0}$ in two dimensions, while showing that the proportion of $b$ -visible points (in such cases) only depends on the negative entries of $b$ .

Keywords

lattice-point visibility, generalized lattice-point visibility, Riemann zeta function

Mathematical Subject Classification 2010

Primary: 11B05

Secondary: 60B10

Milestones

Received: 28 January 2020

Revised: 13 September 2020

Accepted: 28 September 2020

Published: 4 March 2021

Communicated by Stephan Garcia

Authors

Carolina Benedetti
Departamento de Matemáticas Universidad de los Andes Bogotá Colombia

Santiago Estupiñan
Departamento de Matemáticas Universidad de los Andes Bogotá Colombia

Pamela E. Harris
Department of Mathematics and Statistics Williams College Williamstown, MA United States

Vol. 14, No. 1, 2021