Collinear triples, multiplicative subgroups and convex sets

Suppose that we have a finite set of points and lines on the plane. A classical theorem of Szemeredi–Trotter allows us to bound the number of incidences between such points and lines. It turns out that in some problems of additive combinatorics other quantities, namely, the number of collinear points and its analogies, play an important role. Using our method we obtain some applications to the problem of finding lower bounds for the cardinality of subsets of multiplicative subgroups in $Z/pZ$ and also convex subsequences of real numbers.