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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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Yes.

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

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  • $\begingroup$ Note that for unit circles, this may not be best possible, though this bound is current state of the art. See Conjecture 4.2 in the same set of notes. $\endgroup$ Apr 18, 2018 at 0:37

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