Tangent Spheres and Integer Distances

2026-06-16

AI Summary

This study investigates structural constraints on point sets in high-dimensional Euclidean and hyperbolic spaces where all pairwise distances are integers. By analyzing complete bipartite subgraphs in sphere-of-influence graphs and integrating techniques from discrete geometry, graph theory, and hyperbolic geometry, the work extends the classical Erdős–Anning theorem to high-dimensional hyperbolic spaces for the first time. It establishes a quantitative upper bound of $O(D(d+1)^D)$ on the cardinality of such point sets in Euclidean spaces of dimension three or higher. Key contributions include a finiteness criterion for integer-distance point sets, a proof that any such set containing $D+1$ points in general position has size bounded by its diameter and ambient dimension, and the demonstration that in multilateration, $D+1$ non-coplanar beacons suffice to restrict a target’s location to at most two possible positions.

Abstract

The Erdős-Anning theorem states that any point set for which all distances are integers, in a Euclidean space of any dimension, must be either finite or collinear. We prove the same result in hyperbolic space of any dimension. A quantitative form of our result also extends for the first time to Euclidean spaces of dimension greater than two: if a set of points with integer distances in $\mathbb{E}^D$ or $\mathbb{H}^D$ has a subset of $D+1$ points in general position whose diameter is $d$, then the whole set has size $O(D(d+1)^D)$. To prove these results we formulate a lemma that, if the graph of external tangencies of a system of spheres in Euclidean or hyperbolic space contains a $K_{a,b}$ subgraph for $a,b\ge 3$, then the sets of spheres on each side of this biclique have centers that lie on a hyperplane. This lemma also implies that, in multilateration (determining a position from differences of distances to known landmarks), $D+1$ non-coplanar landmarks always suffice to limit the position to two possibilities.

Problem

• integer distances
• Erdős–Anning theorem
• hyperbolic space
• Euclidean space
• point sets

Innovation

• Erdős-Anning theorem
• hyperbolic space
• integer distances
• tangent spheres
• multilateration