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  "path": "/abs/2606.18569v1",
  "publishedAt": "2026-06-18T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "David Eppstein"
  ],
  "textContent": "**Authors:** David Eppstein\n\nThe Erdős-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 $D+1$ points in general position whose diameter is $d$, then the whole set has size $O(D(d+1)^D)$. To prove these results we formulate a lemma that, if the graph of external tangencies of a system of spheres in Euclidean or hyperbolic space contains a $K_{a,b}$ subgraph for $a,b\\ge 3$, then the sets of spheres on each side of this biclique have centers that lie on a hyperplane. This lemma also implies that, in multilateration (determining a position from differences of distances to known landmarks), $D+1$ non-coplanar landmarks always suffice to limit the position to two possibilities.",
  "title": "Tangent Spheres and Integer Distances"
}