Applications of some operators to problems of Number Theory

In the talk a family of operators (finite matrices) with interesting properties will be discussed. This family appeared during attempts to give a simple proof of Chang's theorem from Combinatorial Number Theory. At the moment our operators have found several applications in the area connected with Chang's result as well as another problems of Number Theory as:

• dual Chang's theorem,
• bounds for the additive energy of multiplicative subgroups and convex sets,
• structural results for set with small higher energy,
• estimates of Heilbronn's exponential sums and distribution results of Fermat quotients.