## Abstract

A lattice point $\left(r,s\right)\in {\mathbb{N}}^{2}$ is said to be visible from the origin if no other integer lattice point lies on the line segment joining the origin and $\left(r,s\right)$. It is a well-known result that the proportion of lattice points visible from the origin is given by $1\u2215\zeta \left(2\right)$, where $\zeta \left(s\right)={\sum}_{n=1}^{\infty}1\u2215{n}^{s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika generalized the notion of lattice-point visibility by saying that for a fixed $b\in \mathbb{N}$ a lattice point $\left(r,s\right)\in {\mathbb{N}}^{2}$ is $b$-visible from the origin if no other lattice point lies on the graph of a function $f\left(x\right)=m{x}^{b}$, for some $m\in \mathbb{Q}$, between the origin and $\left(r,s\right)$. In their analysis they establish that for a fixed $b\in \mathbb{N}$ the proportion of $b$-visible lattice points is $1\u2215\zeta \left(b+1\right)$, 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 $b=\left({b}_{1},{b}_{2},\dots ,{b}_{n}\right)\in {\mathbb{N}}^{n}$ the proportion of $b$-visible lattice points is given by $1\u2215\zeta \left({\sum}_{i=1}^{n}{b}_{i}\right)$.

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

$$b=\left({b}_{1}\u2215{a}_{1},{b}_{2}\u2215{a}_{2},\dots ,{b}_{n}\u2215{a}_{n}\right)\in {\left(\mathbb{Q}\setminus \left\{0\right\}\right)}^{n},$$

with nonzero rational entries. In this case, our main result establishes that the proportion of $b$-visible points is $1\u2215\zeta \left({\sum}_{i\in J}\left|{b}_{i}\right|\right)$, where $J$ is the set of the indices $1\le i\le n$ for which ${b}_{i}\u2215{a}_{i}<0$. This result recovers a main theorem of Harris and Omar for $b\in \mathbb{Q}\setminus \left\{0\right\}$ in two dimensions, while showing that the proportion of $b$-visible points (in such cases) only depends on the negative entries of $b$.

## Citation

Carolina Benedetti. Santiago Estupiñan. Pamela E. Harris. "Generalized lattice-point visibility in $\mathbb{N}^k$." Involve 14 (1) 103 - 118, 2021. https://doi.org/10.2140/involve.2021.14.103

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