Lattice point visibility on generalized lines of sight

Abstract

For a fixed $b\in \mathbb{N}=\{1,2,3,\dots \}$ 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 $f(x)=a{x}^b$ 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.

9 pages, 5 figures. To appear in the American Mathematical Monthly

Cited by in corpus (6)

• Generalized Lattice Point Visibility
• Lattice point visibility on power functions
• New methods to find patches of invisible integer lattice points
• The distribution of the generalized greatest common divisor and visibility of lattice points
• Random walks on generalized visible lattice points
• Visible lattice points along curves