Tangent Spheres and Integer Distances (opens in new tab)

The Erd\H{o}s-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 $D+1$ p...