🔢 mathematics

Density of Visible Lattice Points on Hyperplanes and their Intersections

This paper computes the asymptotic density of visible lattice points on hyperplanes and their intersections, deriving a formula involving the Jordan totient function, extending these results to $k$ -th power free points, and characterizing the closure of the set of all such possible densities for a fixed dimension.

Original authors: Finnley Goss, Kelly McKinnie

Published 2026-03-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Finnley Goss, Kelly McKinnie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing at the center of a giant, infinite grid made of integer coordinates (like a massive sheet of graph paper that stretches forever in every direction). Every intersection on this grid is a "lattice point."

Now, imagine you draw a straight line from your position (the origin) to any other point on the grid.

If that line passes through no other grid points before reaching your destination, that point is "visible." It's like looking through a clear window; you have a direct line of sight.
If the line passes through another grid point first, your destination is "hidden" or "invisible" because that first point blocks your view.

Mathematicians have long known that if you look at the entire infinite grid, about 60% of the points are visible (specifically, $1/\zeta(2)$ for a 2D grid). But this paper asks a more specific question: What happens if we only look at points that lie on specific flat surfaces, like walls or floors (called "hyperplanes")?

Here is a breakdown of what Finnley Goss and Kelly McKinnie discovered, using simple analogies.

1. The "Wall" Analogy

Imagine the grid is a 3D room filled with dots.

The Classic View: If you look at the whole room, the density of visible dots is a fixed number.
The New View: Now, imagine you paint a specific wall in the room. This wall is defined by an equation like $2x + 3y + 5z = 10$ . You want to know: Of all the dots sitting exactly on this painted wall, how many are visible from the center?

The authors found a beautiful rule for this. The answer depends entirely on the number on the right side of the equation (in this case, 10).

If the number is 1, every single dot on that wall is visible. (100% visibility).
If the number is a large prime, the visibility drops slightly.
If the number is a composite number (like 10, which is $2 \times 5$ ), the visibility is determined by the "prime ingredients" of that number.

They call this calculation the Jordan Totient Function. Think of it as a "visibility tax." The more prime factors the wall's defining number has, the more "obstacles" (hidden points) you encounter on that specific wall.

2. The Magic Transformation (The "Rubber Sheet")

How did they prove this? They used a mathematical trick called a GL(n, Z) transformation.

The Analogy: Imagine the wall is tilted at a weird, awkward angle. It's hard to count the dots on a tilted, slanted surface.
The Trick: They imagine stretching and skewing the entire room like a rubber sheet. They twist the space until that tilted wall becomes a perfectly flat, vertical wall (like $x = 10$ ).
The Catch: When you stretch the rubber sheet, the grid points get squished and stretched too. However, the authors proved that this stretching does not change the ratio of visible points to hidden points. It's like looking at a photo through a funhouse mirror; the image is distorted, but the percentage of red pixels to blue pixels stays the same.

This allowed them to turn a complex, tilted wall problem into a simple, straight-wall problem that was easy to solve.

3. Intersections: Where Walls Meet

What if you look at the edge where two walls meet? Or the corner where three walls meet?

The Analogy: Imagine looking at the line where the floor meets the wall.
The Result: The same logic applies. The density of visible points on the intersection depends on the "greatest common divisor" of the numbers defining those walls. If the walls are "aligned" in a special way (mathematically speaking), the visibility is high. If they are "misaligned," the visibility drops.

They also generalized this to "k-free" points.

Standard Visible Points: You can't have any common factor (like 2, 3, 5) blocking the view.
Square-Free Points ( $k=2$ ): You only care if a point is blocked by a "square" factor (like 4, 9, 25). It's a looser rule.
Cube-Free Points ( $k=3$ ): You only care about "cube" factors (8, 27).
The paper shows that you can calculate the density for these looser rules too, just by tweaking the formula slightly.

4. The "Gap" Discovery: Not All Densities Are Possible

This is the most surprising part of the paper.

The Question: If you pick any number between 0 and 1, can you find a wall where the visible points have exactly that density?
The Answer (for 2D): Yes. In 2D (flat paper), you can get almost any density. The possible densities are "dense" (like sand grains, you can find one arbitrarily close to any number).
The Answer (for 3D and higher): No. In 3D space (and higher), there are gaps.
- Imagine trying to fill a bucket with water drops of specific sizes. In 2D, the drops are so small you can fill the bucket perfectly. In 3D, the drops are larger. You can fill the bucket, but there will be tiny gaps of air you can never fill.
- The authors proved that for 3D walls, there is a specific "forbidden zone" of densities. You simply cannot find a wall where the visible points make up, say, 45% of the total if that number falls in the gap.

Summary

This paper takes a classic math puzzle about "seeing" points on a grid and solves it for specific slices of that grid (walls and their intersections).

They found a formula: The visibility on a wall depends on the prime factors of the wall's equation.
They used a trick: They twisted space to make hard problems easy, proving the twist doesn't change the answer.
They found a limit: In higher dimensions, not every "percentage of visibility" is possible; there are mathematical "dead zones" where no such wall exists.

It's a story about how geometry, number theory, and a little bit of "rubber sheet" magic combine to reveal the hidden structure of the infinite grid.

1. Problem Statement

The paper addresses the problem of determining the asymptotic density of "visible" lattice points restricted to specific geometric subsets of $\mathbb{Z}^n$ .

Visible Lattice Points: A point $\mathbf{x} = (x_1, \dots, x_n) \in \mathbb{Z}^n$ is visible from the origin if the line segment connecting $\mathbf{x}$ to the origin contains no other lattice points. This is equivalent to the condition $\gcd(x_1, \dots, x_n) = 1$ .
Classic Result: It is well-known that the asymptotic density of visible points in the entire space $\mathbb{Z}^n$ is $1/\zeta(n)$ , where $\zeta$ is the Riemann Zeta function.
The Gap: While the global density is known, the density of visible points restricted to hyperplanes (defined by $\mathbf{a} \cdot \mathbf{x} = b$ ) and their intersections had not been systematically computed in terms of arithmetic functions of the defining parameters.
Generalization: The authors also extend the problem to $k$ -free lattice points, where a point is visible if no prime power $p^k$ divides the greatest common divisor of its coordinates (generalizing the case $k=1$ ).

2. Methodology

The authors employ a combination of analytic number theory, geometry of numbers, and linear algebra over the integers.

Sieve Methods: The core technique involves adapting sieve methods (specifically those found in [18]) to count lattice points coprime to a fixed integer $b$ within a domain. They utilize the identity involving the Möbius function $\mu$ and the Jordan totient function $J_n$ .
Linear Transformations ( $GL_n(\mathbb{Z})$ ): To handle arbitrary hyperplanes, the authors use unimodular transformations (matrices in $GL_n(\mathbb{Z})$ ). Crucially, they prove that these transformations preserve the visibility of lattice points (i.e., $\gcd(\mathbf{x}) = \gcd(A\mathbf{x})$ for $A \in GL_n(\mathbb{Z})$ ).
Smith Normal Form (SNF): For intersections of multiple hyperplanes, the system of linear equations is transformed into a diagonal form using the Smith Normal Form. This reduces the complex intersection of hyperplanes to a simpler coordinate system (e.g., $y_1 = p'_1, \dots, y_m = p'_m$ ), allowing the problem to be reduced to a lower-dimensional visible point density problem.
Volume and Lattice Point Counting: The proofs rely on bounding the error between the volume of a domain and the number of lattice points it contains. They use Lemma 1.2 and Theorem 1.1 from [18], which provide error terms of order $O(\omega^{n-1})$ (where $\omega$ is the diameter) for convex domains.
Achievement Sets (Subsums): In Section 5, the structure of the set of possible densities is analyzed using the theory of "achievement sets" (subsums of convergent series). By taking logarithms, the multiplicative density problem is converted into an additive problem regarding subsums of the series $\sum -\log(1-p^{-n})$ .

3. Key Contributions and Results

A. Density on Single Hyperplanes

The paper establishes a precise formula for the density of visible points on a hyperplane $H$ defined by $\sum a_i x_i = b$ where $\gcd(a_1, \dots, a_n) = 1$ .

Theorem 3.3 & Corollary 3.4: The asymptotic density $\rho(H)$ $ρ (H)$ is given by:
$\rho(H) = \frac{J_{n-1}(b)}{b^{n-1}} = \prod_{p|b} \left(1 - \frac{1}{p^{n-1}}\right)$
where $J_{n-1}$ $J_{n - 1}$ is the Jordan totient function.
- If $b=0$ (the hyperplane passes through the origin), the density is $1/\zeta(n-1)$ .
- If $b=\pm 1$ , the density is $1$ (all points are visible).

B. Density on Intersections of Hyperplanes

The result is generalized to the intersection of $s$ hyperplanes in $\mathbb{R}^n$ .

Theorem 3.3: For a system $A\mathbf{x} = A\mathbf{p}$ with rank $m$ , the density depends on the Smith Normal Form of the coefficient matrix $A$ . Specifically, if $V^{-1}\mathbf{p} = (p'_1, \dots, p'_n)$ , the density is:
$\prod_{p | \gcd(p'_1, \dots, p'_m)} \left(1 - \frac{1}{p^{n-m}}\right)$
This formula accounts for the dimension reduction ( $n-m$ ) and the arithmetic properties of the transformed constant vector.

C. Density of $k$ -Free Points

The authors extend these results to $k$ -free lattice points (where $\gcd(\mathbf{x})$ is not divisible by any $p^k$ ).

Theorem 4.2: The density of $k$ -free points on the intersection of hyperplanes is:
$\prod_{p^k | \gcd(p'_1, \dots, p'_m)} \left(1 - \frac{1}{p^{k(n-m)}}\right)$
This generalizes the standard visible point results (where $k=1$ ).

D. Structure of the Density Set

The paper investigates the set $D_n$ of all possible densities for hyperplanes in $\mathbb{R}^{n+1}$ .

Contrast with $n=1$ : For lines in $\mathbb{R}^2$ ( $n=1$ ), the set of densities is dense in $[0, 1]$ (Schinzel and Sierpiński).
Theorem 5.2: For $n > 1$ , the set of densities $D_n$ is not dense in the interval $[1/\zeta(n), 1]$ . There exist "gaps" in the possible values.
Theorem 5.4: For $n \ge 2$ , $D_n$ is a finite union of compact intervals.
Theorem 5.6: The authors explicitly calculate the structure of $D_2$ (densities of planes in $\mathbb{R}^3$ ), showing it consists of four specific intervals separated by gaps determined by the primes 2 and 3.

4. Significance

Unification of Arithmetic and Geometry: The paper successfully bridges the gap between the geometric distribution of lattice points and the arithmetic properties of the defining equations (via the Jordan totient and Smith Normal Form).
Generalization of Classic Results: It moves beyond the global density of $\mathbb{Z}^n$ to provide exact local densities for constrained subspaces, a problem with applications in Diophantine approximation and cryptography.
Structural Insight: The discovery that the set of possible densities forms a finite union of intervals (rather than being dense or a Cantor set) for dimensions $n > 1$ is a significant structural result in the theory of lattice point distributions.
Methodological Rigor: The use of $GL_n(\mathbb{Z})$ transformations to simplify complex hyperplane intersections while preserving visibility properties provides a robust framework for future research in higher-dimensional lattice problems.

In summary, Goss and McKinnie provide a complete characterization of the asymptotic density of visible and $k$ -free lattice points on hyperplanes and their intersections, revealing that while these densities are determined by simple arithmetic products, the set of all possible densities exhibits a complex, non-dense structure in higher dimensions.