Lattice point visibility on generalized lines of sight

TL;DR

This paper generalizes the concept of lattice point visibility to power functions, establishing the proportion of such points as 1/ζ(b+1), and explores the density and distribution of visible and invisible points for various b.

Contribution

It introduces a new notion of b-visibility on the lattice, derives the proportion of b-visible points as 1/ζ(b+1), and analyzes the existence of large arrays of invisible points for fixed b.

Findings

Proportion of b-visible points is 1/ζ(b+1).

As b increases, the proportion of visible points approaches 1.

Existence of arbitrarily large arrays of b-invisible points for fixed b.

Abstract

For a fixed $b\in\mathbb{N}=\{1,2,3,\ldots\}$ we say that a point $(r,s)$ in the integer lattice $\mathbb{Z} \times \mathbb{Z}$ is $b$-visible from the origin if it lies on the graph of a power function $f(x)=ax^b$ with $a\in\mathbb{Q}$ and no other integer lattice point lies on this curve (i.e., line of sight) between $(0,0)$ and $(r,s)$. We prove that the proportion of $b$-visible integer lattice points is given by $1/\zeta(b+1)$, where $\zeta(s)$ denotes the Riemann zeta function. We also show that even though the proportion of $b$-visible lattice points approaches $1$ as $b$ approaches infinity, there exist arbitrarily large rectangular arrays of $b$-invisible lattice points for any fixed $b$. This work specialized to $b=1$ recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin.