On energies of subsets of combinatoral cubes and varieties

We are going to talk about two questions having in common that in both of them we obtain a series of non–trivial upper bounds for some energies of subsets. The first one is purely combinatoral and we derive an upper bound for the energy of any subset of classical multiplicative / additive combinatoral cube, that is the expression of the form $\sum_{j=1}^d \epsilon_j a_j,$ where $\epsilon_j \in {0,1}$ and $a_j$ are fixed. In the second problem we take an affine or a projective variety $V$ in an algebraic group, $V$ has no large (algebraic) subgroups and obtain that any $A\subseteq V $ enjoys a non-trivial saving for its energy. In both proofs some additive-combinatorial methods are used, there is an application to the restriction problem.  
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000.