Incidences with k-non-degenerate sets and their applications

We study point-sphere and point-plane incidences in three-dimensional space. In particular, for 1 < k < n, we say that a set of spheres (resp., 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 ε > 0, the number of incidences between a set of m points and a k-non-degenerate set of n spheres is