In our talk a family of operators (finite matrices) with interesting properties will be discussed. These operators 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 such as : bounds for the additive energy of some families of sets, new structural results for sets with small higher energy, estimates of Heilbronn's exponential sums and others. Also we discuss some new structural results on sets having critical relations between some kinds of its energies.
Copyright. All Rights Reserved.