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?
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
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
-
$\begingroup$ 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. $\endgroup$ Commented Apr 18, 2018 at 0:37