Let $P$ be a set of $n$ points and let $C$ be a set of $n$ unit circles, both in
$\mathbb{R}^2.$ The maximum number of incidences between P and C is $O(n^{\frac{4}{3}}).$


Is there any bound known for incidences between P and C when $|P| \neq |C|$?



Unbalanced version of incidences between points and unit circles
Let $I(P,C)$ be the number of incidences in your point-circle configuration. One has $$I(P,C) \ll |P|^{2/3} |C|^{2/3} + |C| + |P|,$$ by the [Szemeredi-Trotter theorem.](https://terrytao.wordpress.com/2011/02/18/the-szemeredi-trotter-theorem-via-the-polynomial-ham-sandwich-theorem/) Note that while this is optimal for lines, this is not necessarily optimal for circles. Variants of the grid construction, as in [Chapter 2 of these notes](https://dcg.epfl.ch/files/content/sites/dcg/files/courses/2017%20-%20Additive%20Combinatorics/SumProductBounds_20170529.pdf) are still the best known examples, but they give less than $C\_{\epsilon} n^{1 + \epsilon}$ incidences where $n = |P| = |C|$. For more references, check out the [suggested reading](http://imi.cas.sc.edu/events/nsf-cbms/m2/) for an upcoming conference. 


