Variations on the theme of repeated distances

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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