Density of Visible Lattice Points on Hyperplanes and their Intersections

TL;DR

This paper calculates the asymptotic density of visible lattice points on hyperplanes and their intersections, revealing how these densities relate to number-theoretic functions and extending to power-free points.

Contribution

It provides explicit formulas for the density of visible points on hyperplanes and their intersections, extending classical results to higher dimensions and power-free points.

Findings

Density of visible points on hyperplanes is given by a Jordan totient function ratio.

Extended density formulas to intersections of multiple hyperplanes.

Analyzed the set of all possible densities for fixed dimensions.

Abstract

A lattice point $\vec x=(x_1,\dots,x_n)\in\mathbb Z^{n}$ is said to be visible if the line segment between $\vec x$ and the origin contains no other lattice point. In this paper, we compute the asymptotic density of visible lattice points on hyperplanes and their intersections. In particular, we show that the hyperplane $\vec a \cdot \vec x = b$ in $\mathbb R^{n}$ has visible point density $J_{n-1}(b)/b^{n-1}$ where $J$ is the Jordan totient function. We extend this basic result to find the density of visible points on the intersection of hyperplanes and to the density of $k$-th power free points. Finally, for a fixed dimension $n$, we consider the closure of the set of all possible densities that occur.