Density of Visible Lattice Points on Hyperplanes and their Intersections

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.

Figures (3)

• Figure 1: On the left is an illustration of bounding boxes used in the definition of asymptotic density of visible points in a hyperplane. On the right the picture has been transformed by the matrix $2-110$ taking the line $2x-y=5$ to the line $x=5$ and shows the transformed bounding regions.
• Figure 2: Three lines whose visible point densities successively approximate $\frac{1}{\pi}$ from Example \ref{['ex1']}.
• Figure 3: Blue points are the values of $J_2(b)/b^2$ for all $b$ with primes in its prime factorization up to 61. The color matching vertical lines indicate empty intervals with labels as in the legend. The proof that these are the intervals of $\overline D_2$ is given in Theorem \ref{['D2 intervals']}.

Theorems & Definitions (33)

• Theorem 1.1: Theorem 2.1 of Counting-Problems with $f=1$
• Lemma 1.2
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 3.1
• proof