Variations on the theme of repeated distances

Abstract

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ℝd, d≥4. We also give asymptotically tight upper bounds on the total number of occurrences of the “favourite” distances fromn points in ℝd, d≥4. Related results are proved for distances determined byn disjoint compact convex sets in ℝ2.

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At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.

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Erdős, P., Pach, J. Variations on the theme of repeated distances. Combinatorica 10, 261–269 (1990). https://doi.org/10.1007/BF02122780

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AMS subject classification (1980)

  • 52 A 37
  • 52 A 40