The Erd\H{o}s-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 $\delta$, at most $O(\delta^2)$ 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.

翻译：Erdős-Anning 定理指出，欧几里得平面中任意具有整数距离的点集要么共线，要么有限。更严格地说，对于任意直径为 $\delta$ 的（非退化）三角形，至多有 $O(\delta^2)$ 个点与三角形三个顶点的距离均为整数。我们证明了该结论对平面上任意严格凸距离函数同样成立，并对任意有界亏格的二维完备黎曼流形以及任意三维欧几里得凸集边界的测地距离获得类似结果。作为推论，我们解决了 Richard Guy 于 1983 年提出的关于黎曼流形等边维数的问题。我们的证明基于这些距离的加性加权 Voronoi 图的性质。