  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.
