Incidences with $k$-non-degenerate sets and their applications

We study point-sphere and point-plane incidences in the three-dimensional space. In particular, for $1 < k < n$, we say that a set of spheres (resp., of planes) is $k$-non-degenerate if no circle is contained in $k$ spheres of the set (resp., if no line is contained in $k$ planes of the set). We prove that, for every $\varepsilon>0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ spheres is \[ O(m^{3/4+\varepsilon}n^{3/4}k^{1/4}+n+mk).\] Similarly, we prove that, for every $\varepsilon>0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ planes is \[ O(m^{4/5+\varepsilon}n^{3/5}k^{2/5} + n + mk). \] These bounds are obtained by using the polynomial partitioning technique, recently introduced by Guth and Katz. More specifically, in our proofs we use a pair of constant-degree partitioning polynomials. We also present a couple of applications of $k$-non-degenerate sets: (i) We consider an extension of the three-dimensional unit distances problem, in which we are given a set $D$ of $k$ distinct distances and ask for a three-dimensional set of $m$ points that maximizes the number of pairs of points that span a distance from $D$. By relying on $k$-non-degenerate sets of spheres, we prove an upper bound of $O(m^{236/149+\varepsilon}k^{125/149})$ for the problem (which improves the trivial bound for large values of $k$). Â Â Â (ii) We consider the maximum number of incidences between a three-dimensional set of $n$ planes (without any restrictions) and a set of $m$ points, such that no $k$ points are collinear. Our bound for $k$-non-degenerate planes immediately implies a bound of $O(n^{4/5+\varepsilon}m^{3/5}k^{2/5} + m + nk)$ for this problem, generalizing the previous bound $O(n^{4/5}m^{3/5} + n\log m)$ for the specific case where no three points are collinear (up to the $\varepsilon$ in the exponent).

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