Non-Euclidean ErdÅs-Anning Theorems
For mathematicians studying discrete geometry and distance geometry, this provides a broad generalization of a classical theorem to non-Euclidean settings, resolving an open problem.
The paper extends the Erdős-Anning theorem (integer-distance point sets are collinear or finite) to strictly convex distance functions on the plane, two-dimensional complete Riemannian manifolds of bounded genus, and geodesic distance on boundaries of 3D convex sets, resolving a 1983 question by Richard Guy on equilateral dimension of Riemannian manifolds.
The ErdÅs-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.