Nondegenerate spheres in four dimensions
Non-degeneracy was first defined for hyperplanes by Elekes-T\'oth, and later extended to spheres by Apfelbaum-Sharir: given a set $P$ of $m$ points in $\mathbb{R}^d$ and some $\beta\in(0,1)$, a $(d-1)$-dimensional sphere (or a $(d-1)$-sphere) $S$ in $\mathbb{R}^d$ is called $\beta$-nondegenerate with respect to $P$ if $S$ does not contain a proper subsphere $S'$ such that $|S'\cap P|\geq \beta|S\cap P|$. Apfelbaum-Sharir found an upper bound for the number of incidences between points and nondegenerate spheres in three dimensions, which was recently used by Zahl to obtain the best known bound for the unit distance problem in three dimensions. In this paper, we show that the number of incidences between $m$ points and $n$ $\beta$-nondegenerate 3-spheres in $\mathbb{R}^4$ is $O_{\beta,\varepsilon}\left(m^{\frac{15}{19}+\varepsilon} n^{\frac{16}{19}}+mn^{\frac{2}{3}}\right)$. As a consequence, we obtain a bound of $O_{\varepsilon}(n^{2+4/11+\varepsilon})$ on the number of similar triangles formed by $n$ points in $\mathbb{R}^4$, an improvement over the previously best known bound $O(n^{2+2/5})$. While proving this, we find it convenient to work with a more general definition of nondegeneracy: a bipartite graph $G=(P,Q)$ is called $\beta$-nondegenerate if $|N(q_1)\cap N(q_2)|<\beta |N(q_1)|$ for any two distinct vertices $q_1,q_2\in Q$; here $N(q)$ denotes the set of neighbors of $q$ and $\beta$ is some positive constant less than 1. A $\beta$-nondegenerate graph can have up to $\Theta(|P||Q|)$ edges without any restriction, but must have much fewer edges if the graph is semi-algebraic or has bounded VC-dimension. We show that previous results for planes and spheres in three dimensions still hold under this new definition, and so does our new bound for spheres in four dimensions.