OpenAI research collection · 1 manuscript · Source checked 2026-10-19
For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝ^(d) determine at least c_dn^{2/d} distinct distances, with c_d> 0 depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
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For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝ^(d) determines at least c_d n^{2/d} distinct distances, where c_d> 0 depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.
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