Energies and structure of additive sets

In the paper we prove that any sumset or difference set has large E_3 energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms. In particular, we give criteria for a set to be a 1) set of the form H+L, where H+H is small and L has"random structure", 2) set equals a disjoint union of sets H_j, each H_j has small doubling, 3) set having large subset A' with 2A' is equal to a set with small doubling and |A'+A'| \approx |A|^4 / \E(A).


Introduction
Let G = (G, +) be an abelian group. For two sets A, B ⊆ G define the sumset as A + B := {x ∈ G : x = a + b , a ∈ A , b ∈ B} and, similarly, the difference set Also denote the additive energy of a set A by E(A) = E 2 (A) = |{a 1 − a ′ 1 = a 2 − a ′ 2 : a 1 , a ′ 1 , a 2 , a ′ 2 ∈ A}| , and E k (A) energy as The special case k = 1 gives us E 1 (A) = |A| 2 because of there is no any restriction in the set from (1). So, the cardinality of a set can be considered as a degenerate sort of energy. Note that a trivial upper bound for E k (A) is |A| k+1 . Now recall a well-known Balog-Szemerédi-Gowers Theorem [37].
Theorem 1 Let A ⊆ G be a set, and K ≥ 1 be a real number. Suppose that E(A) ≥ |A| 3 /K. Then there is A ′ ⊆ A such that |A ′ | ≫ |A|/K C and where C > 0 is an absolute constant.
So, Balog-Szemerédi-Gowers Theorem can be considered as a result about the structure of sets A having the extremal (in terms of its cardinality or E 1 (A) in other words) value of E(A). Namely, any of such a set has a subset A ′ with the extremal value of the cardinality of its difference set. These sets A ′ are called sets with small doubling. On the other hand, it is easy to obtain, using the Cauchy-Schwarz inequality, that any A having subset A ′ such that (2) holds, automatically has polynomially large energy E(A) ≫ K |A| 3 (see e.g. [37]). Moreover, the structure of sets with small doubling is known more or less thanks to a well-known Freiman's theorem (see [37] or [25]). Thus, Theorem 1 finds subsets of A with rather rigid structure and, actually, it is a criterium for a set A to be a set with large (in terms of E 1 (A)) the additive energy: E(A) ∼ K |A| 3 ∼ K (E 1 (A)) 3/2 .
In the paper we consider another extremal relations between different energies and describe the structure of sets having these critical relations. Such kind of theorems have plenty of applications. It is obvious for Balog-Szemerédi-Gowers theorem, see e.g. [37], [3], [14], [15], [19], [21] and so on; for recent applications using critical relations between energies E 2 (A) and E 3 (A), see e.g. [32], [33] and others. Before formulate our main results let us recall a beautiful theorem of Bateman-Katz [3], [4] which is another example of theorems are called "structural" by us.
Here T 4 (A) is the number of solutions of the equation a 1 + a 2 + a 3 + a 4 = a ′ 1 + a ′ 2 + a ′ 3 + a ′ 4 , a 1 , a 2 , a 3 , a 4 , a ′ 1 , a ′ 2 , a ′ 3 , a ′ 4 ∈ A and this characteristic is another sort of energy. One can check that any set satisfying (3)-(5), E(A) = |A| 2+τ 0 and all another conditions of the theorem is an example of a set having T 4 (A) ≪ |A| 4+3τ 0 +σ 0 . Note that if E(A) = |A| 2+τ 0 then by the Hölder inequality one has T 4 (A) ≥ |A| 4+3τ 0 . Thus, Theorem 2 gives us a full description of sets having critical relations between a pair of two energies: E(A) and T 4 (A).
There are two opposite extremal cases in Theorem 2 : α = 0 and α = 1−τ 0 2 . For simplicity consider the situation when G = F n 2 . In the case α = 0 by Bateman-Katz result our set A, roughly speaking, is close to a set of the form H ∔ Λ, where ∔ means the direct sum, H ⊆ F n 2 is a subspace, and Λ ⊆ F n 2 is a dissociated set (basis), |Λ| ∼ |A|/|H| ∼ |A| 1−τ 0 . These sets are interesting in its own right being counterexamples in many problems of additive combinatorics. The reason for this is that they have mixed properties : on the one hand they contain translations H + λ, λ ∈ Λ of really structured set H but on the other hand they have also some random properties, for example, its Fourier coefficients (see the definition in section 2) are small. Our first result says that a set A is close to a set of the form H ∔ Λ iff there is the critical relation between E 3 (A) and E(A), that is E 3 (A) ≫ |A|E(A), more precisely, see Theorem 22. In the situation α = 1−τ 0 2 , G = F n 2 our set A looks like a union of (additively) disjoint subspaces H 1 , . . . , H k (see example (iii) from [26]) with k = |A| 1+τ 0 2 . Such sets can be called self-dual sets, see [33]. In our second result we show that, roughly speaking, any such a set has critical relation between E 3 (A) (more precisely E(A) · E 4 (A)) and so-called Gowers U 3 -norm of the set A (see the definition in section 7) and vice versa. Theorem 46 contains the exact formulation.
These two structural results on sets having critical relations between a pair of its energies are the hearth of our paper. In the opposite of Theorem 2 almost all bounds of the paper are polynomial, excluding, of course, the dependence on the number k of the considered energies E k , T k or U k if its appear. Moreover the first structural theorem hints us a partial answer to the following important question. Consider the difference set D = A − A or the sumset S = A + A of an arbitrary set A. What can we say nontrivial about the energies of D, S in terms of the energies of A? In view of the first of the main examples above, that is G = F n 2 , A = H ∔ Λ, |Λ| = K, E(A) ∼ |A| 3 /K we cannot hope to obtain a nontrivial bound for the additive energy of D or S because in the case D = S = H ∔ (Λ + Λ), and so it has a similar structure to A with Λ replacing by Λ + Λ. On the other hand, we know that the sets of the form H ∔ Λ have large E 3 energy. Thus, one can hope to obtain a good lower bound for E 3 (D) and E 3 (S). It turns out to be the case and we prove it in section 6. Roughly speaking, our result asserts that if |D| = K|A|, E(A) ≪ |A| 3 /K then E 3 (D) ≫ K 7/4 |A| 4 , and a similar inequality for A + A. The paper is organized as follows. We start with definitions and notations used in the paper. In the next section we give several characterisations of sets of the form A = H ∔ Λ, where H is a set with small doubling, Λ is a "dissociated" set. Also we consider a "dual" question on sets having critical relations between T 4 and E energies, that is the situation when T 4 (A) is large in terms on E(A). It was proved that, roughly, A contains a large subset A ′ such that the sequence A ′ , 2A ′ , 3A ′ , . . . is stabilized at the second step, namely, A ′ + A ′ is a set with small doubling and, besides, |A ′ + A ′ | ≈ |A| 4 /E(A) in the only case when T 4 (A) ≫ |A| 2 E(A), see Theorem 23.
Section 6 contains the proof of inequality (6) and we make some preliminaries to this in section 5. For example, we obtain in the section an interesting characterisation of sumsets S = A + A or difference sets D = A − A with extremal cardinalities of intersections and, similarly, for S, see Theorem 28. It turns out that for such sets D, S the set A should have either very small O(|A| k+o(1) ) the energy E k (A) or very large ≫ |A| 3−o(1) the additive energy. In other words either A has "random behaviour" or, in contrary, is very structured. Clearly, both situations are realized: the first one in the situation when A is a fair random set (and hence A ± A has almost no structure) and the second one if A is a set with small doubling, say.
In section 7 we consider some simple properties of Gowers norms of the characteristic function of a set A and prove a preliminary result on the connection of E(A) with E(A∩ (A+ s)), s ∈ A−A, see Theorem 39. It gives a partial counterexample to a famous construction of Gowers [14], [15] of uniform sets with non-uniform intersections E(A ∩ (A + s)) (see the definitions in [14], [15] or [37]). We show that although all sets A ∩ (A + s) can be non-uniform but there is always s = 0 such that E(A ∩ (A + s)) ≪ |A ∩ (A + s)| 3−c , c > 0, provided by some weak conditions take place. This question was asked to the author by T. Schoen. In the next section we develop the investigation from the previous one and characterize all sets with critical relation between Gowers U 3 -norm and the energies E, E 4 or E 3 . Also we consider some questions on finding in A a family of disjoint sets A ∩ (A + s) or its large disjoint subsets.
A lot of results of the paper such as Bateman-Katz theorem are proved under some regular conditions on A. For example, the assumption from Theorem 2 require that for all A * ⊆ A, |A * | ≫ |A| the following holds E(A * ) ≫ E(A). We call the conditions as connectedness of our set A (see the definitions from sections 3, 7) and prove in the appendix that any set contains some large connected subset. Basically, we generalize the method from [31].
Thus, we have characterized two extremal situations of Theorem 2 in terms of energies. Is there some similar characterisation for other cases of the result? Do exist criteria in terms of energies for another families of sets? Finally, are there further characteristics of sumsets/difference sets which separate it from arbitrary sets?
The author is grateful to Vsevolod F. Lev and Tomasz Schoen for useful discussions.

Definitions
Let G be an abelian group. If G is finite then denote by N the cardinality of G. It is wellknown [23] that the dual group G is isomorphic to G in the case. Let f be a function from G to C. We denote the Fourier transform of f by f , where e(x) = e 2πix and ξ is a homomorphism from G to R/Z acting as ξ : x → ξ · x. We rely on the following basic identities and If where for a function f : The k-fold convolution, k ∈ N we denote by * k , so * k := * ( * k−1 ). We use in the paper the same letter to denote a set S ⊆ G and its characteristic function S : G → {0, 1}. Clearly, S is the characteristic function of a set iff Write E(A, B) for the additive energy of two sets A, B ⊆ G (see e.g. [37]), that is and by (9), Let T k (A) := |{a 1 + · · · + a k = a ′ 1 + · · · + a ′ k : a 1 , . . . , a k , a ′ 1 , . . . , a ′ and more generally Let also σ k (A) := (A * k A)(0) = |{a 1 + · · · + a k = 0 : a 1 , . . . , a k ∈ A}| .
Notice that for a symmetric set A that is A = −A one has σ 2 (A) = |A| and σ 2k (A) = T k (A). and be the higher energies of A and B. The second formulas in (15), (16) can be considered as the definitions of E k (A), E k (A, B) for non integer k, k ≥ 1. Similarly, we write E k (f, g) for any complex functions f , g and more generally . where We also put ∆(x) = ∆({x}), x ∈ G.
Quantities E k (A, B) can be written in terms of generalized convolutions.
Quantities E k (A) and T k (A) are "dual" in some sense. For example in [33], Note 6.6 (see also [28]) it was proved that provided by k is even. Moreover, from (7)-(10), (12) For a positive integer n, we set [n] = {1, . . . , n}. Let x be a vector. By x denote the number of components of x. All logarithms are to base 2. Signs ≪ and ≫ are the usual Vinogradov's symbols and if the bounds depend on some parameter M polynomially then we write ≪ M , ≫ M . If for two numbers a, b the following holds a ≪ M b, b ≪ M a then we write a ∼ M b. In particular, a ∼ b means a ≪ b and b ≪ a.
All polynomial bounds in the paper can be obtained in explicit way.
Lemma 4 Let A ⊆ G be a set. Then for all positive integers n, m the following holds We need in several quantitative versions of the Balog-Szemerédi-Gowers Theorem. The first symmetric variant is due to T. Schoen [27].
Also we need in a version of Balog-Szemerédi-Gowers theorem in the asymmetric form, see [37], Theorem 2.35.
Theorem 6 Let A, B ⊆ G be two sets, |B| ≤ |A|, and M ≥ 1 be a real number. Let also L = |A|/|B| and ε ∈ (0, 1] be a real parameter. Suppose that Then there are two sets H ⊆ G, Λ ⊆ G and z ∈ G such that and The next lemma is a special case of Lemma 2.8 from [34]. In particular, it gives us a connection between E 3 (A) and E(A, A s ), see e.g [28].
Lemma 7 Let A ⊆ G be a set. Then for every k, l ∈ N s,t: In particular, Now recall a lemma from [30], [33]. and Let also give a simple Corollary 18 from [32].
We give a small generalization of Proposition 11 from [28], see also [21].
Our first lemma from [33] (where some operators were used in the proof) is about a nontrivial lower bound for E s (A), s ∈ [1,2] in terms of E(A).
Lemma 12 Let A ⊆ G be a set, and β, γ ∈ [0, 1]. Suppose that A is (2, β, γ)-connected with β ≤ 1/2. Then for any s ∈ [1, 2] the following holds The second lemma from [33] is about an upper bound for eigenvalues of some operators. To avoid of using the operators notation we formulate the result in the following way.
Lemma 13 Let A ⊆ G be a set. Then for an arbitrary function f : A → C one has Further, there is a set A ′ ⊆ A, |A ′ | ≥ |A|/2, namely, such that for any function f : A ′ → C the following holds Moreover for any even real function g there is a set A ′ ⊆ A, |A ′ | ≥ |A|/2 such that for any function f : A ′ → C the following holds Note that for the characteristic functions f of sets from A bound (31) can be obtained using the Cauchy-Schwarz inequality. Further, estimate (34) is a generalization of (33) which was proved in [33], see Lemma 44. Bound (34) can be obtained in a similar way.
We finish the section noting a generalization of formula (23) of Lemma 8. That is just a part of Lemma 4.2 from [33].

Lemma 14
Let A, B ⊆ G be finite sets, S ⊆ G be a set such that A + B ⊆ S. Suppose that ψ is a function on G. Then

Structural results
In this section we obtain several general structural results, some of which have applications to sum-products phenomenon, for example. These results are closely related to the Balog-Szemerédi-Gowers Theorem, and we adopt the convention of writing G as an additive group. The proofs follow the arguments from [28] and [29]. Now we formulate the first result of the section.
Proposition 15 Let A ⊆ G be a finite set, and M ≥ 1, η ∈ (0, 1] be real numbers. Let E(A) = |A| 3 /K. Suppose that for some set P ⊆ A − A the following holds and Then for any ε ∈ (0, 1), there are two sets H ⊆ G, Λ ⊆ G and z ∈ G such that and P r o o f. Using Lemma 8 with P * = P , we see that Note that s : |As|< Applying Lemma 7, that is the formula E 3 (A) = s E(A, A s ), combining with (41), we get Put µ := max Using (42), we have Applying the asymmetric version of Balog-Szemerédi-Gowers Theorem 6, we find two sets Λ, H such that (38)-(40) take place. This completes the proof. ✷ We write the fact that sets A, H, Λ satisfy (38)-(40) with η ≫ 1 as Note that the degree of polynomial dependence in formula (43) is a function on ε.
Example 16 Let H ⊆ F n 2 be a subspace and Λ ⊆ F n 2 be a dissociated set (basis). Put A = H ∔Λ, where ∔ means the direct sum, and |Λ| = K. Detailed discussion of the example can be found, e.g. in [33]. If s ∈ H then A s = A and hence A + A s = A + A. If s ∈ (A + A) \ H then A s is the disjoint union of two shifts of H and thus |A + A s | ≤ 2|A|. Whence and E(A) ∼ |A| 3 /K. It means that condition (37) takes place in the case A = H ∔ Λ.
Taking P = A − A and applying Proposition 15 as well as formulas (26), (27) of Lemma 10, we obtain the following consequence.
Corollary 17 Let A ⊆ G be a set, M ∈ R, ε ∈ (0, 1) and E(A) = |A| 3 /K. Then either Note that for any set A ⊆ G with E(A) = |A| 3 /K the inequality |A 2 ± ∆(A)| ≥ K|A| 2 follows from Lemma 8 and a trivial estimate E 3 (A) ≤ |A|E(A). We will deal with the reverse condition E 3 (A) ≫ |A|E(A) in Proposition 20 and Theorem 21 below.
The next corollary shows that if a set A is not close to a set of the form Λ ∔ H then there is some imbalance (in view of Plünnecke-Ruzsa inequality (18)) between the doubling constant and the additive energy of A or A − A.
and the Cauchy-Schwarz inequality, we get Hence where the assumption of the corollary has been used. This completes the proof. ✷ The quantities E(A ± A) (and hence |A 2 ± ∆(A)| in view of Lemma 10, see also Proposition 29 below) appear in sum-products results (in multiplicative form). For example, in [21] the following theorem was proved.
Here E × (A) := |{a 1 a 2 = a 3 a 4 : a 1 , a 2 , a 3 , a 4 ∈ A}|. Thus, by the obtained results, we have, roughly, that either E × (A), E × (AA) can be estimated better then by Lemma 10, formulas (26), (27) or A has the rigid structure A ≈ Λ · H. Usually the last case is easy to deal with. Similar methods were used in [21]. Proposition 20 Let A ⊆ G be a set, and M ≥ 1 be a real number. Suppose that and Further, take any ε ∈ (0, 1) and put K : Put P = P j and ∆ = 2 j |A|/(2 2 M ). Thus Hence, by the Cauchy-Schwarz inequality Note that and therefore It follows that and so there exists x ∈ A such that the set Finally, from (48), say, one has and because of , note that by the first inequality of (49) and the bound |P | ≤ 2 6 Also, by the definition of the number K, and inequality (53) the following holds Applying the asymmetric version of Balog-Szemerédi-Gowers Theorem 6 with A = A, B = P , and recalling (53), we obtain the required inequalities, excepting the first inequality of (38), Thus, by the definition of the number ∆ and estimate (53), we obtain Hence there is w ∈ A such that This completes the proof. ✷

Assumption (45) of the Proposition 20 is a generalisation of the usual condition
Theorem 21 Let A ⊆ G be a set, s ≥ 3 be a positive integer, and M ≥ 1 be a real number. Suppose that Take any ε ∈ (0, 1) and put K : P r o o f. The arguments almost repeat the proof of Proposition 20, so we skip some details. Using dyadic pigeonholing and the assumption, we find Thus, by the Cauchy-Schwarz inequality On the other hand |P |∆ s−1 ≤ E s−1 (A) and hence Note that by (56) and our choice of the set P , we have and, again, After that apply the asymmetric version of Balog-Szemerédi-Gowers Theorem 6 and an analog of the arguments from (54)-(55). This concludes the proof. ✷ The more general assumption Thus, we have considered the common case. Note, finally, that estimates (46), (47) are the best possible. Indeed, take G = F n 2 , A = H ∔ Λ, where H ≤ F n 2 is a linear subspace and Λ is a dissociated set (basis). Now we can prove a criterium for sets having critical relation between E(A) and E 3 (A) energies.
In other words To get the last estimate we have used the fact |H| ≫ M,K ε E(A)|A| −2 . This completes the proof. ✷

Recall that
We conclude the section proving a "dual" analogue of Proposition 20, that is we replace the condition on E 3 (A) with a similar condition for T 4 (A) and moreover for T s (A). Again, the proof follows the arguments from [28].
Theorem 23 Let A ⊆ G be a set, and M ≥ 1 be a real number. Suppose that and s ≥ 2 then formulas (58), (59) take place. Conversely, bounds (58) where and using assumptions (57), (60), combining with the definition of sets P j , Ω, we have Applying the Hölder inequality, we get Put P = P j , f = f j and ∆ = 2 j T s /(2 4 M a s ). Thus Clearly, |P | ≤ 4T s ∆ −2 . Using the last inequality, the definition of the number ∆ and bound (63), we obtain To estimate the size of P we note by (63) that After that use arguments (51)-(52) of the proof of Proposition 20. By Theorem 5 there is for every n, m ∈ N. We have By (64) and the definition of the number ∆ there is x ∈ (s−1)A such that the set A ′ := A∩(P ′ −x) has the size at least We have by (65) that Conversely, applying bound (59) with n = m = s, combining with the Cauchy-Schwarz inequality, we obtain Using (58), we get In other words, T 2s (A) ≫ M, log |A|, s |A| 2s T s (A). This completes the proof. ✷ So, we have proved in Theorem 23 that, roughly speaking, A ′ − A ′ is a set with small (in terms of the parameter M ) doubling and vice versa. Note, that we need in multiple |A| 3 E −1 (A) in (59), because by (58) and the Cauchy-Schwarz inequality, we have the same lower bound for |A ′ − A ′ |. Thus, A ′ does not equal to a set with small doubling but A ′ − A ′ does. Results of such a sort were obtained in [28], [32] and [33].
It is easy to see that an analog of Theorem 23 takes place if one replace (57) onto condition T s (A) ≥ |A| 2(s−2) E(A)/M , where s is an even number, s ≥ 4, and, further, even more general relations between T k energies can be reduced to the last case and Theorem 23 via a trivial estimate T s (A) ≤ |A| 2 T s−1 (A). We do not need in such generalizations in the paper.

Sumsets: preliminaries
Let A ⊆ G be a set. Before studying the energies of sumsets or difference sets we concentrate on a following related question, which was asked to the author by Tomasz Schoen. Namely, what can be proved nontrivial concerning lower bounds for |A ± A s |, s = 0? The connection with A ± A is obvious in view of Katz-Koester trick (28). We start with a result in the direction.
, and a = |A|. Let us begin with (66). Denote by ω the maximum in (66). By the Cauchy-Schwarz inequality and formula (31) of Lemma 13, we obtain Multiplying the last inequality by |A s |, summing over s = 0 and using the assumption On the other hand, by Lemma 8, we have Combining (69) with (70), we obtain Now let us obtain (67). Using Lemma 13, we find A ′ , |A ′ | ≥ |A|/2 such that estimate (33) takes place. As in (68), we get By assumption E 3 (A) ≥ 2 4 γ −1 |A| 3 . Using the connectedness, we obtain Multiplying inequality (71) by |A ′ s |, summing over s = 0, we have in view of (72) that Combining the last formula with first inequality from (72), we get On the other hand, summing the first estimate from (71) over s = 0 and applying Lemma 7, we see that Combining (74), (75), we obtain This completes the proof. ✷ It improves a trivial lower bound |A ± A s | ≥ |A|. Using bound (67) one can show that there is , provided by some connectedness assumptions take place.
We need in lower bounds on E k (A) in Theorem 24 and Proposition 25 below to be separated from a very natural simple example, which can be called a "random sumset" case. Namely, take a random A ⊆ G and consider A ± A. This "random sumset" has almost no structure (provided by A ± A is not a whole group, of course) and we cannot say something useful in the situation. It does not contradict Theorem 24 and Proposition 25 (see also the results of the next section) because the energies E k (A) are really small in the case. Now we give another prove of estimate (67) which can be derived from inequality (77), case k = 2 below. Actually, it gives us even stronger inequality, namely, |A 2 − ∆(A s )| ≫ |A| 5 E −1 (A) for some s = 0, or, more generally, (see formulas (79), (80) below) provided by some connectedness assumptions take place. In particular, taking r = k and r = 1 in the previous formula, we get Proposition 25 Let A ⊆ G be a set, k ≥ 2 be a positive integer. Take two sets D, S such that P r o o f. Using Lemma 13, we find A ′ , |A ′ | ≥ |A|/2 such that estimate (34) takes place with Multiplying the last inequality by |A ′ x | k−1 and summing over x = 0 (to obtain (76) multiply by |A ′ x | r−1 ), we get Here we have used the fact and the assumption E k+1 (A) ≥ 2 k+2 γ −1 |A| k+1 . Note that by Katz-Koester trick (29), one has and similarly for S. Some weaker results but without any conditions on A were obtained in [28].
Results above give an interesting corollary on non-random sumsets/difference sets. Put D = A−A, S = A+A. Then for an arbitrary positive integer k and any elements a 1 , . . . , a k ∈ A, we have In particular, there is x ∈ D, x = 0 such that |D x |, |S x | ≥ |A| (also it follows from Katz-Koester inclusion (29)). By Corollary 17 (see also Lemma 10) one can improve it to |D x |, |S x | ≥ K ε |A|, where K = |A| 3 E −1 (A). Theorem 24 gives us even stronger result.
One can get an analog of Corollary 26 for multiple intersections (81) but another types of energies will require in the case. Nevertheless, some weaker inequality of the form |D x | ≥ K ε |A| can be obtained, using Proposition 20 and Theorem 21. Here K = |A| k+1 E −1 k (A), k ≥ 2. Interestingly, we do not even need in any connectedness in this weaker result.
where the constant c K ε satisfies c K ε ≫ K ε 1 and, again, the degree of the polynomial dependence is a function on c.
P r o o f. Suppose not. Take any set P ⊆ A k−1 − ∆(A). Applying Lemma 7, one has Note that the assumption where the sum ′ above is taken over x = (x 1 , . . . , x k−1 ) with distinct x j . Now take P such that ∆ < |A x | ≤ 2∆ for x = (x 1 , . . . , x k−1 ) ∈ P, where all x j , j ∈ [k] are distinct and Of course, such P exists by the pigeonhole principle and bound (84). Using the last inequality, and recalling (83), we obtain In other words, In particular, there are at least We can suppose that the summation in (85) is taken over x = (x 1 , . . . , x k−1 ) with distinct x j because of the rest is bounded by The last estimate follows from the assumption K ≤ |A| 1−c . Choosing any such x and using the Cauchy-Schwarz inequality, we obtain and in view of Katz-Koester trick (29), we see that |D x | , |S x | are huge for large K 1 . This concludes the proof. ✷ Using Proposition 27 one can derive an interesting dichotomy.
Suppose that for any vector x = (x 1 , . . . , x k−1 ) with distinct x j , j ∈ [k] the following holds Then either E k (A) ≪ M, |A| ε , k |A| k or E(A) ≫ M, |A| ε , k |A| 3 . Again, the degree of the polynomial dependence is a function on ε.
Finally, by the upper bound for the parameter K the number c which is defined as K = |A| 1−c can by taken depends on ε only. Thus, everything follows from Proposition 27, the only thing we need to consider is the situation when E k (A) ≤ k 2 E k−1 (A). But in the case In other words E(A) ≫ k |A| 3 and this concludes the proof. ✷ Thus, if |D x | , |S x | are not much larger than |A| then either A is close to what we called a "random sumset" or, on the contrary, is very structured. Clearly, the both situations are realized.

Energies of sumsets
Let A ⊆ G be a set. Throughout the section we put D = A−A and S = A+A. As was explained in the introduction that one can hope to prove a good lower bound for E 3 (D). It will be done in Theorem 30 below but before this we formulate a simple preliminary lower bound for E D k (D), E D k (S). Similar lower bounds for E D 2 (D), E D 2 (S) were given in Corollary 5.6 of paper [21]. Further, it was proved in [28] and, similarly, P r o o f. The second estimates in (86), (87) follow from Lemma 10. Further, it is easy to get (or see e.g. [28]) that as required. Similarly, Finally, by Lemma 10, we get This completes the proof. ✷ Interestingly, that some sort of sumset, namely, A n ± ∆(A) gives a lower bound for an energy, although, usually, the energy provides lower bounds for the cardinality sumsets via the Cauchy-Schwarz inequality. The trick allows to obtain a series of results in [28]- [30]. Although bounds (86), (87) are very simple they can be tight in some cases. For example, take A to be a dissociated set or, in contrary, a very structural set as a subspace. Now we formulate the main result of the section concerning lower bounds for some energies of sumsets/difference sets. Again we need in lower bounds on E k (A) in Theorem 30 below to be separated from the "random sumset" case.
, and a = |A|. Let us obtain bounds (88), (90). Using Lemma 13, we find A ′ , |A ′ | ≥ |A|/2 such that estimate (33) takes place. As in the proof of inequalities (68), (71), we get Multiplying the last inequality by |A ′ s |, summing over s and using Katz-Koester trick, we have and similar for S. On the other hand by the second part of Lemma 8, we obtain and using the first part of the lemma, we have the same bound for S [30]). Another way to prove the same is to use Lemma 14 with A = B = A ′ , ψ(x) = (A ′ • A ′ )(x). Combining (93) and (94), (95), we get Using the tensor trick (see [37] or [33]), we have (88). If A is (2, β, γ)-connected then

E(A ′ ) ≫ γE(A)
and combining the last inequality with (93) and the second bound from (94), we get (90) (to obtain lower bound for E 2 3 (S, A) one should use Lemma 14). It remains to prove (89) and (91). Returning to (92)-(93), we obtain or, by the Hölder inequality On the other hand, by Lemma 8 for any P ⊆ A ′ − A ′ , we have Applying the Hölder inequality, we get and similarly for S. Now choose P ⊆ A ′ − A ′ such that P = {s ∈ A ′ − A ′ : ρ < |A ′ s | ≤ 2ρ} for some positive number ρ and such that s∈P Of course, the set P exists by Dirichlet principle. Combining the last inequality with (97), we obtain Using estimates (96), (98), we have Thus Applying the tensor trick again, we get (89). To obtain (91) recall that A is (3/2, β, γ)-connected set. Hence by (99), we obtain and the first formula of (91) follows. Further, because of A is (2, β, γ)-connected set then using Lemma 12 for A ′ as well as (99), we have An upper bound here is K 2 |A| 4 and it follows from the main example of section 4, that is G = F n 2 , A = H ∔ Λ. Note also that the second inequality in formula (91) is weak but do not depends on the size of A − A or on the energy E 3/2 (A).
As for dual quantities T k (D), T k (S), our example A = H ∔ Λ shows that there are not nontrivial lower bounds for T k (A ± A) in general, which is better than a simple consequence of Katz-Koester (just use (D • D)(x) ≥ |A|D(x) and (S • S)(x) ≥ |A|D(x)). The reason is that the structure of A ± A is similar to the structure of A in the case, of course. Nevertheless, it was proved in [28], and also in [33], Note 6.6 that where P ⊆ D is any set. So, if we know something on E 2k (A) then it gives us a new information about T k (A ± A). Trivially, formula (101) implies that T k (A ± A) ≥ |A| 2k+2 /E(A). Again, the last inequality is sharp as our main example A = H ∔ Λ shows.
Vsevolod F. Lev asked the author about an analog of (102) for different sets A and B. Proposition 31 below is our result in the direction. The proof is in spirit of [33]. For simplicity we consider the case k = 2 only. The case of greater powers of two is considered similarly if one take M 2 , M 4 , . . . or just see the proof of Theorem 6.3 from [33] (the case of any k). We do not insert the full proof because we avoid to use the operators from [32], [33] in the paper which is considered to be elementary. and calculate its rectangular norm where λ j (M ) the singular numbers of M . Clearly, Thus, by the Cauchy-Schwarz inequality, we get as required. This completes the proof. ✷ We end this section showing that there is a different way to prove our Theorem 30 using slightly bigger sets D x or S x not A ± A x . The proof based on a lemma, which demonstrates, in particular, that A ± A contains approximately |A| 3 E −1 (A) almost disjoint translates of A, roughly. In the proof we use arguments from [1].
Lemma 32 Let A, B ⊆ G be two sets. Then there are P r o o f. Let us begin with (103). Put S = A + B. Our arguments is a sort of an algorithm. At the first step of the algorithm take Suppose that our algorithm stops after s steps. If s ≥ |B|/2 then we are done. Put U = s j=1 A j and B * = B \ {b 1 , . . . , b s }. Then s|A|/2 < |U | ≤ s|A| and |B * | ≥ |B|/2. We have  4εσ/(ab)). Thus, Finally, we obtain and hence, in view of the condition σ ≥ 16b the following holds A, B) .
provided by (3, β, γ)-connectedness assumptions, β, γ ≫ 1 (by the way bound (105) is tight as our main example H ∔ Λ shows). On the other hand, we have by Lemma 8 that Combining the last two bounds, we get (90).
Using similar arguments and the second part of Lemma 32, we obtain the following consequence, which shows, in particular, that the popular difference sets [15], [37] have some structure in the sense that they have large energy of some sort.
P r o o f. Let σ = σ P (A). On the one hand, using Lemma 32 with A = A, B = −A, we construct the family of disjoint sets P j ⊆ P ∩ (A − a j ), a j ∈ A, |P j | ≥ 2 −3 σ|A| −1 , the number s satisfies (104). Put A j = P j + a j ⊆ A. Thus, by connectedness of our set A, we have On the other hand, applying Lemma 8, we get Combining the last two inequalities, we obtain bound (106). This concludes the proof. ✷
Definition 34 Gowers U d -norm (or d-uniformity norm) of the function f is the following expression A sequence of 2 d points x ω ∈ G d , ω ∈ {0, 1} d is called d-dimensional cube in G d or just a d-dimensional cube. Thus the summation in formula (107) is taken over all cubes of G d . For example, {(x, y), (x ′ , y), (x, y ′ ), (x ′ , y ′ )}, where x, x ′ , y, y ′ ∈ G is a two-dimensional cube in G×G. In the case Gowers norm is called rectangular norm.
For d = 1 the expression above gives a semi-norm but for d ≥ 2 Gowers norm is a norm. In particular, the triangle inequality holds [15] f One can prove also (see [15]) the following monotonicity relation.
for all d ≥ 2.
If G = (G, +) is a finite Abelian group with additive group operation +, N = |G| then one can "project" the norm above onto the group G and obtain the ordinary ("one-dimensional") Gowers norm. In other words, we put the function f (x 1 , . . . , x d ) in formula (107) equals "onedimensional" function f (x 1 , . . . , x d ) := f (pr(x 1 , . . . , x d )), where pr(x 1 , . . . , x d ) = x 1 + · · · + x d . Denoting the obtained norm as U d , we have an analog of (109), see [15], [37] for all d ≥ 2. It is convenient to write In the case f = A, where A ⊆ G is a set, we have by formula (112) that where π(s 1 , . . . , s d ) is a vector with 2 d components, namely,
In definitions (107), (111) we have used the size of the set G/group G. The results of the paper are local, in the sense that they do not use the cardinality of the container group G. Thus it is natural to ask about the possibility to obtain an analog of (110), say, without any N in the definition. That is our simple result in the direction.

Proposition 35
Let A ⊆ G be a set. Then for any integer k ≥ 2 one has In particular, P r o o f. We have Thus, if the summation in (116) is taken over the set then it gives us (1 − 1/2k) proportion of the norm A U k . Let us estimate the size of Q k . Clearly, Certainly, the same bound holds for the cardinality of any set of tuples (s i 1 , . . . , s i k−1 ) defined similar to (117) having the size k − 1. Hence, by the projection results, see e.g. [5], we see that the summation in (116) is taken over a set S of vectors (s 1 , . . . , s k ) of size at most (2k . Returning to (116) and using the Cauchy-Schwarz inequality as well as formula (113), we obtain

The last inequality implies that
where 0 < C k < 1 depends on k only. Using the tensor trick we obtain the result. This completes the proof. ✷ Remark 36 Estimate (114) is sharp as an example of a sufficiently dense random subset of a group G shows. For higher Gowers norms one can show by induction a similar sharp inequality . It demonstrates expected exponential (in terms of E(A)) growth of the norms.
In the next section we will need in a statement, which is generalizes lower bound for U 3norm (115).
Lemma 37 Let A, B ⊆ G be two sets. Then P r o o f. We use the same arguments as in the proof of Proposition 35. One has Because of s |A s | = |A| 2 , s |B s | = |B| 2 , we get for the set S above that |S| ≪ (|A| 2 |B| 2 E −1 (A, B)) 2 . Thus and the result follows. ✷ Using (115) and the Cauchy-Schwarz inequality, we have a consequence.

Corollary 38
Let A ⊆ G be a set and |A − A| ≤ K|A| or |A + A| ≤ K|A|. Then Inequality (115) gives us a relation between A U 3 = s E(A s ) and E(A). W.T. Gowers (see [15]) constructed a set A having a random behavior in terms of E(A) (more precisely, he constructed a uniform set, that is having small Fourier coefficients, see [15]) such that for all s the sets A s have non-random (non-uniform) behavior in terms of E(A s ). Nevertheless, it is natural to ask about the possibility to find an s = 0 with a weaker notion of randomness, that is E(A s ) ≪ |A s | 3−c , c > 0. This question was asked to the author by T. Schoen. We give an affirmative answer on it.
Theorem 39 Let A ⊆ G be a set, E(A) = |A| 3 /K. Suppose that for all s = 0 the following holds where M ≥ 1 is a real number. Let K 4 ≤ M |A|. Then there is s = 0 such that |A s | ≥ |A|/2K and  Clearly,C (x, y) ≤ C(x, y) ≤ min{|A x |, |A y |} .
We will write P x := P ∩ ({x} × G), and P y := P ∩ (G × {y}). Put also Our first lemma says that the size of P and some characteristics of the set can be estimated in terms of L and M .
Further, for any nonzero y and λ the following holds P r o o f. By the Cauchy-Schwarz inequality, we have We can assume that |A| ≥ 4KL 1/2 because otherwise the result is trivial in view of the condition it follows by (121) and the assumption |A| ≥ 4KL 1/2 that In other words |A| 2 4LM 2 ≤ |P|. On the other hand and we obtain the required upper bound for the size of P. Further, for any fixed y = 0 the following holds Finally, Now, we show that some norm of P is huge. Actually, we use the function C notC in the proof.
Lemma 41 One has P r o o f. As in the proof of Lemma 40, we get x,y P(x, y)A(z + x)A(z + y) .
Using the Cauchy-Schwarz inequality, we obtain Applying the Cauchy-Schwarz inequality again, we have This concludes the proof of the lemma. ✷ In terms of the sets P λ we can rewrite expression (124) as In view of estimate (124), a trivial inequality and bounds for size of P λ of Lemma 40 the sets P λ should look, firstly, like some sets with small doubling (more precisely as sets with large additive energy) and, secondly, the large proportion of such sets must correlate to each other. A model example here is P λ (x) = Q(x + α(λ)), where α is an arbitrary function and Q is an arithmetic progression of size approximately Θ(|A|/K). Now we prove that the example is the only case, in some sense.
Put l = log(LM ). Note that l ≤ log(KM ) because if L ≥ K then the result is trivial. We can suppose that and a calculation 2(4KL|A|) 2 4L|A| 2 ≤ |A| 5 2 10 KL 4 M 6 . Below we will assume that any summation is taken over nonzero indices λ, µ. By Lemma 40 and a trivial estimate where ∆ * = |A| 2 16 KL 6 M 6 . Using the pigeonhole principle and Lemma 40, we find a number ∆ such that ∆ * ≤ ∆ ≤ 4M 2 L|A| K and From (127) and the Cauchy-Schwarz inequality one can see that the summation in the formula is taken over Put Q = P µ . We have E(Q) ≥ ε|Q| 3 . Applying a trivial general bound we get by Lemma 40 Given an arbitrary λ let the maximum in the last formula is attained at point x := α(λ). Thus, we have Hence we find a set Q of the required form, that is having the large additive energy and which is correlates with sets P λ . Now, we transform the obtained information into some knowledge about the original set A.
Using the definition of the set P, we obtain We know that E(Q) ≥ ε|Q| 3 . By Balog-Szemerédi-Gowers Theorem 5 we find Q ′ ⊆ Q, |Q ′ | ≫ ε|Q| such that |Q ′ − Q ′ | ≪ ε −4 |Q ′ |. We will prove shortly that the set Q in (129) can be replaced by a setQ, namely where c(ε) > 0 is some constant depends on L and M only andQ has small doubling. Indeed, starting with (128), put Q ′ 1 = Q ′ , and define inductively disjoint sets then we stop the algorithm. Let the procedure works exactly s steps and put B = B s ,B = Q\B.
We claim that s ≪ ε −1 . To prove this note that if (131) does not hold then In other words, E(B j ) ≫ ε|Q| 3 and, hence, It means, in particular, that after s ≪ ε −1 number of steps our algorithm stops indeed. At the last step, we get by the construction that B,B)) .
Let us prove the following estimate If not then by the Cauchy-Schwarz inequality and the choice ofB, we obtain and we get a contradiction. Hence the following holds Applying the Hölder inequality, we find a set Q ′ j such that So, putting Q ′ := Q ′ j we get (130) with c(ε) ≫ ζε 2 lL 2 M 2 . Of course, the summation in the obtained formula can be taken just over λ with |A λ | ≫ c(ε) |A| K and we will assume this.
Denote by Ω the set From (130) and our assumption (119) we have |Ω| ≫ c(ε)M −1 |A| 2 . On the other hand, considering Ω λ := {z : (z, λ) ∈ Ω} for any fixed λ, one has Hence there are at least ≫ c 2 (ε)KM −4 L −1 |A| sets A λ such that there exists some z = z(λ) with (Q • A λ )(z − α(λ)) ≫ c(ε)|A|/2K. Denote the set of these λ by T . For any such A λ there exists a shift of the setQ such that We have by Lemma 4 that for any λ 1 , λ 2 ∈ T the following holds In particular, Finally, using (125) as well as Lemma 7 with k = l = 2, we obtain with the required lower bound for L. This completes the proof of Theorem 39. ✷ We finish the section by analog of Definition 11, which we will use in the next section.
Definition 42 For β, γ ∈ [0, 1] a set A is called U k (β, γ)-connected if for any B ⊆ A, |B| ≥ β|A| the following holds Again, if, say, γ −1 |A| 8 /|A ± A| 4 ≥ A U 3 then by inequality (115) one can see that A is U 3 (β, γ)-connected for any β. The existence of U k (β, γ)-connected subsets in an arbitrary set is discussed in the Appendix.

Self-dual sets
Inequality (115) gives us a relation between A U 3 and E(A). It attaints at a random subset A of G, where by randomness we mean that each element of A belongs to the set with probability E(A)/|A| 3 . On the other hand, it is easy to see that an upper bound takes place A weaker estimate follows from (132) combining with the Cauchy-Schwarz inequality In the section we consider sets having critical relations between A U 3 and E 4 (A), E(A) that is the sets satisfying the reverse inequality to (133) (actually, we use a slightly stronger estimate then reverse to (132)). It turns out that they are exactly which we called self-dual sets.
Let us recall a result on large deviations. The following variant can be found in [7].
We need in a combinatorial lemma.
Lemma 44 Let ∆, σ, C > 1 are positive numbers, t be a positive integer, and M 1 , . . . , M t be sets, Then there are at least P r o o f. We will choose our setsM i randomly with probability at least 1/4. First of all, we note that Put p = t∆2 −1 σ −1 . In view of (134), we get p ∈ (0, 1/2]. Let us form a new family of sets taking a set M i from M 1 , . . . , M t uniformly and independently with probability p. Denote the obtained family as M ′ 1 , . . . , M ′ s . By Lemma 43 and bound (134), we have after some calculations that with probability at least 3/4. Further the expectation of σ equals by our choice of p. Hence, by Markov inequality, with probability at least 1/2 one has and by the Cauchy-Schwarz inequality, we get Finally, we putM i = M ′′ i . By our choice of parameters and estimates (135) the following holds .
Similarly, the number n of the setsM i can be estimated from (136) Of course, such set P ′ exists by the pigeonhole principle. To apply Lemma 44, we need to calculate the quantity σ By the last identity and estimate (33) (for details, see [33]), we get Applying Lemma 44, we find disjoint In the last inequality we have used (2, β, γ)-connectedness of A. To complete the proof note that Clearly, the bound on k in Corollary 45 is the best possible up to logarithms. Calculating E(A, A j )/|A j | and comparing its with E 3 (see [33]) one can obtain an alternative proof of lower bounds for |A ± A s | as of section 5. Another result on a family with disjoint A s is proved in Proposition 49 below. Now we are able to obtain the main result of the section.
Theorem 46 Let A ⊆ G be a set, and M ≥ 1 be a real number. Put l = log |A|. Suppose that A is U 3 (β, γ) and (2, β, γ)-connected with β ≤ 0.5. Then inequality takes place iff there is a positive real ∆ ∼ M, l E 3 (A)E(A) −1 and a set such that |P | ≫ M, l |A|, P = −P , further, and such that for any s ∈ P there is H s ⊆ A s , |H s | ≫ M, l ∆, with and E(A, H s ) ≪ M, l |H s | 3 . Moreover there are disjoint sets H j ⊆ A s j , |H j | ≫ M, l ∆, s j ∈ P , j ∈ [k] such that all H j have small doubling property (139), E(A, H j ) ≪ M, l |H j | 3 and k ≫ M, l |A|∆ −1 .
. Let us begin with the necessary condition. Using Lemma 13, we find A ′ , |A ′ | ≥ |A|/2 such that estimate (33) takes place. Because of A is U 3 (β, γ) and (2, β, γ)-connected with β ≤ 0.5, we have A ′ U 3 ∼ A U 3 and E(A ′ ) ∼ E(A). Combining assumption (137) with the Cauchy-Schwarz inequality, we get In particular, by the last inequality and (132), (133), we obtain . With some abuse of the notation we will use the same letter A for A ′ below. By Lemma 7, we have One can assume that the summation in the last formula is taken over s such that Returning to (140), we obtain So, we know all energies E, E 3 , E 4 if we know |P | and ∆. Let us estimate the size of the P . Taking any s ∈ P , we get by Lemma 13 that and k ≫ |P | 2 ∆σ −1 . Arguing as in Corollary 45, we get σ ≪ E|P ||A| −1 and hence k ≫ |P |∆|A|E −1 ≫ M, l |A|∆ −1 . Of course the last bound on k is the best possible up to constants depending on M , l. We have obtained the necessary condition. Let us prove the sufficient condition. Using the Cauchy-Schwarz inequality and formulas (138)-(139), we have as required. This completes the proof. ✷

Remark 47
In the statement of Theorem 46 there is the set of popular differences P and the structure of A is described in terms of the set P . Although, we have obtained a criterium it can be named as a weak structural result. Perhaps, a stronger version avoiding using of the set P takes place, namely, under the hypothesis of Theorem 46 there are disjoint sets H j ⊆ A s j , It is easy to see that it is a sufficient condition. Indeed, because of the sets H j ⊆ A are disjoint, we have Using the assumption |H j − H j | ≪ M, l |H j |, the first bound from (143), as well as Corollary 38, we obtain and, similarly, by the second and the third inequality of (143), we get as required.
Example 48 Let A ⊆ G be a set having small Wiener norm, that is the following quantity Using the Hölder inequality twice (see also [19]), we get Thus, an application of Theorem 46 gives us that A has very explicit structure (2-connectedness follows from (144) and U 3 -connectedness can be obtained via formula (118) in a similar way). Another structural result on sets from F p with small Wiener norm was given in [19].
If Theorem 39 does not hold that is E(A s ) ≫ |A s | 3 for all s then, clearly, A U 3 ≫ E 3 (A) and we can try to apply our structural Theorem 46. On the other hand, if A is a self-dual set, that is a disjoint union of sets with small doubling then for any s = 0 one has exactly E(A s ) ≫ |A s | 3 . It does not contradict to Theorem 39 because of condition (119).
Roughly speaking, in the proof of Theorem 46 we found disjoint subsets of A s , containing huge amount of the energy (see also Corollary 45). One can ask about the possibility to find some number of disjoint A s (and not its subsets) in general situation. Our next statement answer the question affirmatively.
Proposition 49 Let A ⊆ G be a set, D ⊆ A − A. Put Then there are at least l ≥ |D| 2 /(4σ) disjoint sets A s 1 , . . . , A s l . In particular, if |A − A s | ≤ σ we find s 2 ∈ D 1 such that Put D 2 = D 1 \ (A − A s 2 ). If |D 1 | < |D|/2 then terminate the algorithm. And so on. At the last step, we obtain the set D l = D \ l j=1 (A − A s j ), |D l | < |D|/2. It follows that Thus l ≥ |D| 2 /(4σ). Finally, recall that Thus all constructed sets A s 1 , . . . , A s l are disjoint.
To get (146) put D = A− A and recall that by Lemma 10 the following holds |A 2 − ∆(A)| = s∈A−A |A − A s |. This completes the proof. ✷ One can ask is it true that not only E 3 energy but U 3 -norm of sumsets or difference sets is large? It is easy to see that the answer is no, because of our basic example A = H ∔ Λ, |Λ| = K. In the case E(A) ∼ |A| 3 /K, E 3 (A) ∼ |A| 4 /K but A U 3 ∼ |A| 4 /K 2 and similar for A ± A.

Appendix
In the section we prove that any set contains a relatively large connected subset. The case k = 2 of Proposition 55 below was proved in [31] (with slightly worse constants) and we begin with a wide generalization.
Definition 50 Let X, Y be two nonempty sets, |X| = |Y |. A nonnegative symmetric function q(x, y), x ∈ X, y ∈ Y is called weight if the correspondent matrix q(x, y) is nonnegatively defined.
Having two sets A and B put E q (A, B) := x,y q(x, y)A(x)B(y), E q (A) := E q (A, A). Clearly, E q (A, B) ≤ |A||B| q ∞ . The main property of any weight is the following.
Lemma 51 Let q be a weight. Then for any sets A, B, one has Example 52 Clearly, the function q(x, y) = (B • B) k (x − y) for any set B and an arbitrary positive integer k is a weight. Further, by the construction of Gowers U d -norms it follows that is also a weight for any nonnegative function f . In formula (148), we have x = (x 1 , . . . , x d ), x ′ = (x ′ 1 , . . . , x ′ d ), pr(y 1 , . . . , y d ) := y 1 + · · · + y d . Another example of a weight is where f is an arbitrary nonnegative function again and h = (h 1 , . . . , h d−1 ).
For two sets S, T ⊆ G, S = ∅, T ⊆ S put µ S (T ) = |T |/|S|. Now we prove a general lemma on connected sets and quantities E q , where q is a weight.
After that applying the same arguments to the set A 1 , find a subset C ⊆ A 1 such that (150) does not hold (if it exists) and so on. We obtain a sequence of sets A ⊇ A 1 ⊇ · · · ⊇ A s , and |A s | ≤ (1 − β 1 ) s |A|. So, at the step s, we have Thus, our algorithm must stop after at most s ≤ log(1/c)(2 log( 1−β 2 ρ 1−β 1 )) −1 number of steps. Putting A ′ = A s , we see that inequality (150) takes place for anyÃ ⊆ A ′ with β 1 |A ′ | ≤ |Ã| ≤ β 2 |A ′ |. Finally, by the second estimate in (152), we obtain (151). This concludes the proof. ✷ Let us formulate a useful particular case of Lemma 53.