We study higher moments of convolutions of the characteristic function of a set, which generalize a classical notion of the additive energy. Such quantities appear in many problems of additive combinatorics as well as in number theory. In our investigation we use different approaches including basic combinatorics, Fourier analysis and eigenvalues method to establish basic properties of higher energies. We provide also a sequence of applications of higher energies additive combinatorics.

A set of reals A = {a 1 , . . . , a n } is called convex if a i+1 − a i > a i − a i−1 for all i. We prove, among other results, that for some c > 0 every convex A satisfies |A − A| c|A| 8/5 log −2/5 |A|.

We prove that if for a finite set A of integers we have |A + A| ≤ K|A|, then A is contained in a generalized arithmetic progression of dimension at most K 1+C(log K) −1/2 and size at most exp(K 1+C(log K) −1/2)|A| for some absolute constant C. We also discuss a number of applications of this result.

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If $k$ is a positive integer, we say that a set $A$ of positive integers is $k$-sum-free if there do not exist $a,b,c$ in $A$ such that $a + b = kc$. In particular we give a precise characterization of the structure of maximum sized $k$-sum-free sets in $\{1,\ldots,n\}$ for $k\ge 4$ and $n$ large.