[Paper Review] Bounds for the minimum diameter of integral point sets

Sascha Kurz, Reinhard Laue|arXiv (Cornell University)|Apr 8, 2008

Mathematical Approximation and Integration12 references3 citations

TL;DR

This paper establishes new upper bounds for the minimum diameter $d(m,n)$ of integral point sets in $m$-dimensional Euclidean space, where all pairwise distances are integers. By constructing point sets using geometric configurations—such as points on a line with one off-line point, or points on spheres around a line—it proves $d(m,m^2+m) \leq 17$ and provides exact values for $d(2,n)$ up to $n=122$, including $d(3,24)=244$, significantly improving known bounds.

ABSTRACT

Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $\mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $\mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the smallest possible diameter $d(m,n)$ for given parameters $m$ and $n$. We give some new upper bounds for the minimum diameter $d(m,n)$ and some exact values.

Motivation & Objective

• To determine tighter upper bounds for the minimum diameter $d(m,n)$ of integral point sets in $m$-dimensional Euclidean space.
• To identify exact values of $d(m,n)$ for specific $m$ and $n$, especially in low dimensions.
• To develop geometric constructions that yield integral point sets with small diameter, enabling improved bounds.
• To investigate the asymptotic behavior and structural constraints of minimal diameter integral point sets.

Proposed method

• Constructing $m$-dimensional integral point sets by placing $n-1$ points on a line and one point at height $h$ from the line, using known planar integral point sets.
• Extending planar configurations to higher dimensions by placing $n'$ points on an $(m-1)$-dimensional sphere of radius $h$ centered on the line, ensuring all pairwise distances remain integral.
• Using Pythagorean triples to ensure integral distances between the off-line point and points on the line, particularly by solving $f^2 + v^2 = w^2$ with $w$ integer.
• Applying truncation of regular simplices to generate $m^2 + m$-point integral point sets with controlled diameter.
• Exhaustive enumeration via computational methods to verify exact values of $d(2,n)$ for $n$ up to 122.
• Proving tightness of bounds through structural analysis and comparison with known configurations, including the use of symmetric and regular arrangements.

Experimental results

Research questions

• RQ1 What is the smallest possible diameter $d(m,n)$ for an $m$-dimensional integral point set of size $n$?
• RQ2 Can new upper bounds for $d(m,n)$ be derived using geometric constructions that preserve integrality of all pairwise distances?
• RQ3 What exact values of $d(m,n)$ can be determined for specific $m$ and $n$, particularly in low dimensions?
• RQ4 How do the bounds for $d(m,n)$ behave asymptotically as $n$ increases for fixed $m$?
• RQ5 Is the bound $d(m,m^2+m) \leq 17$ tight, and can it be generalized to other configurations?

Key findings

• The paper establishes $d(m,m^2+m) \leq 17$ for all $m \geq 2$, providing a significant improvement over previous bounds for this class of configurations.
• Exact values of $d(2,n)$ are computed for $n$ from 90 to 122, with $d(2,122) = 4883$, extending the known sequence of minimal diameters.
• The value $d(3,24) = 244$ is established as the exact minimum diameter for a 3-dimensional integral point set of size 24.
• The construction via truncated regular simplices yields an $m$-dimensional integral point set with $m^2 + m$ points and diameter 17 when $a=7$, $b=8$, proving the bound $d(m,m^2+m) \leq 17$.
• The bound $d(m,n-2+m) \leq d(2,n)$ is proven for $9 \leq n \leq 122$, showing that planar minimal configurations can be lifted to higher dimensions with controlled diameter.
• The paper confirms that $d(3,9) = 16$, correcting a prior literature error that listed it as 17.