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.

We obtain a generalization of the recent Kelley-Meka result on sets avoiding arithmetic progressions of length three. In our proof we develop the theory of the higher energies. Also, we discuss the case of longer arithmetic progressions, as well as a general family of norms, which includes the higher energies norms and Gowers norms. * 8l lk 1

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|.

Annotation.We improve a result of Solymosi on sum-products in R, namely, we prove that max {|A + A|, |AA|} ≫ |A| 4 3 +c , where c > 0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets A ⊂ R with |AA| ≤ |A| 4/3 .

Let F be a field with positive odd characteristic p. We prove a variety of new sum-product type estimates over F . They are derived from the theorem that the number of incidences between m points and n planes in the projective three-spacewhere k denotes the maximum number of collinear planes.The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely thatfor any A such that |A| < p 5 8 .for the sumset A + A of A ⊆ F , and similarly for the product set AA, alias A · A. Sometimes we write nA for multiple sumsets, e.g. A+A+A = 3A, as well as A −1 = {a −1 : a ∈ A\{0}}.

We improve a previous sum-products estimates in R, namely, we obtain that max {|A + A|, |AA|} ≫ |A| 4 3 +c , where c any number less than 5 9813 . New lower bounds for sums of sets with small the product set are found. Also we prove some pure energy sum-products results, improving a result of Balog and Wooley, in particular.

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We prove a range of new sum-product type growth estimates over a general field F, in particular the special case F = F p . They are unified by the theme of "breaking the 3/2 threshold", epitomising the previous state of the art.This concerns two pivotal for the sum-product theory questions, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of F, as well as the number of values of the symplectic form determined by a finite subset of F 2 .We establish the estimate |R[A]| |A| 8/5 for cardinality of the set R[A] of distinct cross-ratios, defined by triples of elements of a (sufficiently small if F has positive characteristic, similarly for the rest of the estimates) set A ⊂ F, pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set P ⊂ F 2 , the number of distinct values of the symplectic form ω on pairs of distinct points u, u ′ of P is |ω(P )| |P | 2/3+c , with an explicit c. Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein.Our proofs strongly use specially suited applications of new incidence bounds over F, which apply to higher moments of representation functions. The technical thrust of the paper is using additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any F, by showing that if A is sufficiently small and has additive doubling constant M , then * |AA| M −2 |A| 14/9 . This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set A× A ⊂ F 2 , the quantity often arising in applications of geometric incidence estimates. (Expansion for large sets) IfF = F p , then |R[A]| ≫ min p, |A| 5/2 p 1/2 . In particular, if |A| ≥ p 3/5 , then |R[A]| ≫ p. 3. (Uniform expansion) If F = F p , then |R[A]| ≫ min{p, |A| 3 2 + 1 22 log − 4 9 |A|}.