Given $n$ points and an integer $k ≥ 2.$ What is the maximum number of unit circles which pass through at least $k$ of the points?
I think the answer is $O(n^{4/3}/k),$ but I'm not really sure. Any ideas?
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Szemerédi-Trotter can be extended to unit circles. See, for example, Theorem 4.1 here: http://math.caltech.edu/~2014-15/3term/ma191c-sec2/1%20Classic%20DG.pdf