Counting Arcs in .

An arc in is a set such that no three points of are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let denote the family of all arcs in .

Convexity, superquadratic growth, and dot products.

Let be a point set with cardinality . We give an improved bound for the number of dot products determined by , proving that 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 , there exist such that 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.