
ВИДЕОТЕКА 

An elementary approach to the operator method in additive combinatorics К. И. Ольмезов^{} 

Аннотация: The additive energy $$ a+b=c+d, a, b, c, d \in A. $$ Finding the upper bounds for the additive energy of sets from some given classes is very popular subject of additive combinatorics. In 2012 Shkredov provided operator method that allows to estimate $$ E_3(A):=\# \{a_1b_1=a_2b_2=a_3b_3:a_i, b_i \in A, i=1,2,3\} $$ and the common energy $$ E(A,D):=\# \{a_1d_1=a_2d_2:a_i \in A, d_i \in D, i=1,2\} $$ for an arbitrary $$ T_A(x,y)=(A \circ A)(xy)=\#\{ab=xy:a,b \in A \} $$ for obtain a certain very general inequality to the number of the solutions of two linear equation systems. We suggest an elementary approach to prove this inequality which gives an elementary proof of the corresponding bounds of Also we discuss a generalization of this inequality (which is elementary as well) connecting with the Sidorenko's conjecture from graph theory. Sidorenko's conjecture states that for any bipartite graph $$ t_H \le t_{K_2} (G), $$ where We consider a special case of the graph 