  Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an
improved bound for the number of dot products determined by $P$, proving that,
\[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the
proof of this bound is a new superquadratic expander involving products and
shifts. We prove that, for any finite set $X \subset \mathbb R$, there exist
$z,z' \in X$ such that \[
\left|\frac{(zX+1)^{(2)}(z'X+1)^{(2)}}{(zX+1)^{(2)}(z'X+1)}\right| \gtrsim
|X|^{5/2}. \] This is derived from a more general result concerning growth of
sets defined via convexity and sum sets, and which can be used to prove several
other expanders with better than quadratic growth. The proof develops arguments
from recent work by the first two listed authors and Misha Rudnev, and uses
predominantly elementary methods.
