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. Our proofs are based on the properties of additively weighted Voronoi diagrams of these distances.

翻译：Erdős–Anning定理指出，欧几里得平面中所有距离均为整数的点集要么共线，要么有限。更强地，对于任意直径为$\delta$的（非退化）三角形，至多$O(\delta^2)$个点能与该三角形的三个顶点保持整数距离。我们证明了该结论对平面上任意严格凸的距离函数同样成立，并对任意有界亏格二维完备黎曼流形以及任意三维欧几里得凸集边界上的测地距离得到了类似结果。我们的证明基于这些距离的加权重Voronoi图的性质。