There exist n-point planar sets where every subset A has a distance occurring ≥|A|²/n^{1−δ} times, confirming Erdős's 1980 isosceles-triangle conjecture and answering a repeated-distance question negatively.
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The Minkowski grid has robustly many repeated distances
There exist n-point planar sets where every subset A has a distance occurring ≥|A|²/n^{1−δ} times, confirming Erdős's 1980 isosceles-triangle conjecture and answering a repeated-distance question negatively.