A new upper bound is shown for the number of incidences between n points and n families of concentric circles in the plane. As a consequence, it is shown that the number of the k most frequent distances among n points in the plane is f n (k)=O(n 1.4571 k .6286 ) improving on an earlier bound of Akutsu, Tamaki, and Tokuyama.
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Solymosi, Tardos & Tóth The k Most Frequent Distances in the Plane. Discrete Comput Geom 28, 639–648 (2002). https://doi.org/10.1007/s00454-002-2896-z