Non-Euclidean Erdős-Anning Theorems

Abstract

The Erdős-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$δ$, at most $O(δ^2)$ points can have integer distances from all three triangle vertices. We prove the same results for any strictly convex distance function on the plane, and analogous results for every two-dimensional complete Riemannian manifold of bounded genus and for geodesic distance on the boundary of every three-dimensional Euclidean convex set. As a consequence, we resolve a 1983 question of Richard Guy on the equilateral dimension of Riemannian manifolds. Our proofs are based on the properties of additively weighted Voronoi diagrams of these distances.

Figures (13)

• Figure 1: Three star-shaped sets, each shown with a point in its kernel.
• Figure 2: Integer $L_1$ (left) and $L_\infty$ distances in the integer grid (\ref{['ex:grid-distances']}). The yellow shaded regions are the unit balls for these two distances.
• Figure 3: Construction for an infinite non-geodesic set with integer distances for a convex but not strictly convex distance function $d_K$ (\ref{['ex:non-geodesic']}).
• Figure 4: The points on the unit sphere whose angles are even multiples of the angles of a 3-4-5 right triangle have pairwise rational distances.
• Figure 5: An additively weighted Voronoi diagram with two degenerate sites (red and green) and two non-degenerate sites (yellow and blue). The red site has equal weighted distance to itself and to the yellow site; its Voronoi cell (shown as a red line segment) extends along a ray, away from the yellow site, to the boundary of the yellow Voronoi cell. The green site has larger weighted distance to itself than to the blue site, so its Voronoi cell is empty. The Voronoi cells for the yellow and blue sites together cover the plane, separated from each other by the central vertical line. The distance function is unspecified; it could be Euclidean, or any other strictly convex distance function whose unit disk has vertical reflection symmetry (so that the bisector of the two horizontally-aligned yellow and blue points is a vertical line).

Theorems & Definitions (81)

• Example 1
• Example 2
• Example 3
• Example 4
• Lemma 2
• proof
• Definition 1
• Lemma 3
• proof
• Corollary 4