January 24, Wednesday, 15:30, room 211

Sofia Aleshina

On some problems of sum-product type

Let A be a finite non-empty set of elements of a ring. Consider the set A+A of pairwise sums and the set A*A of pairwise products of the elements in A. Obviously, the cardinality of such sets is at least |A|. The Erdős–Szemerédi hypothesis says that max(|A+A|,|A*A|) ≫ C|A|^(2-ɛ), where ɛ and C are positive constants. That is, either A+A or A*A is quite large. In this talk, I will say a few words about the Erdős–Szemerédi theorem and other results over Z and Z/pZ. Later I consider a generalization of the problem of sum-product type. Namely, I consider the set of values f(x,y) that a homogeneous polynomial f in two variables takes if x and y belong to A. I obtain a lower bound on the cardinality of this set.