On iterated product sets with shifts II
aa r X i v : . [ m a t h . N T ] J un ON ITERATED PRODUCT SETS WITH SHIFTS II
BRANDON HANSON, OLIVER ROCHE-NEWTON, AND DMITRII ZHELEZOV
Abstract.
The main result of this paper is the following: for all b ∈ Z there exists k = k ( b ) suchthat max {| A ( k ) | , | ( A + u ) ( k ) |} ≥ | A | b , for any finite A ⊂ Q and any non-zero u ∈ Q . Here, | A ( k ) | denotes the k -fold product set { a · · · a k : a , . . . , a k ∈ A } .Furthermore, our method of proof also gives the following l ∞ sum-product estimate. For all γ > C = C ( γ ) such that for any A ⊂ Q with | AA | ≤ K | A | and any c , c ∈ Q \ { } , there are at most K C | A | γ solutions to c x + c y = 1 , ( x, y ) ∈ A × A. In particular, this result gives a strong bound when K = | A | ǫ , provided that ǫ > Introduction
Background and statement of main results.
Let A be a finite set of rational numbersand let u ∈ Q be non-zero. In this article we wish to investigate the sizes of the k -fold product sets A ( k ) := { a · · · a k : a , . . . , a k ∈ A } and ( A + u ) ( k ) = { ( a + u ) · · · ( a k + u ) : a , . . . , a k ∈ A } . This is an instance of a sum-product problem. Recall that the Erd˝os-Szemer´edi [7] sum-productconjecture states that, for all ǫ > c ( ǫ ) > {| A + A | , | AA |} ≥ c ( ε ) | A | − ε holds for any A ⊂ Z . Here A + A := { a + b : a, b ∈ A } is the sum set of A , and AA is anothernotation for A (2) . Erd˝os and Szemer´edi also made the more general conjecture that for any finite A ⊂ Z , max {| kA | , | A k |} ≥ c ( ǫ ) | A | k − ǫ , where kA := { a + · · · + a k : a , . . . , a k ∈ A } is the k -fold sum set . Both of these conjectures arewide open, and it is natural to also consider them for the case when A is a subset of R or indeedother fields. The case when k = 2 has attracted the most interest. See, for example, [12], [13],[16], [17] and the references contained therein for more background on the original Erd˝os-Szemer´edisum-product problem.Most relevant to our problem is the case of general (large) k . Little is known about the Erd˝os-Szemer´edi conjecture in this setting, with the exception of the remarkable series of work of Chang [6] and Bourgain-Chang [4]. This culminated in the main theorem of [4]: for all b ∈ R there exists k = k ( b ) ∈ Z such that max {| kA | , | A k |} ≥ | A | b (1)holds for any A ⊂ Q . On the other hand, it appears that we are not close to proving such a strongresult for A ⊂ R .In the same spirit as the Erd˝os-Szemer´edi conjecture, it is expected that an additive shift willdestroy multiplicative structure present in A . In particular, one expects that, for a non-zero u , atleast one of | A ( k ) | or | ( A + u ) ( k ) | is large. The k = 2 version of this problem was considered in [9]and [11]. The main result of this paper is the following analogue of the Bourgain-Chang Theorem. Theorem 1.1.
For all b ∈ Z , there exists k = k ( b ) such that for any finite set A ⊂ Q and anynon-zero rational u , max {| A k | , | ( A + u ) k |} ≥ | A | b . This paper is a sequel to [10], in which the main result was the following.
Theorem 1.2.
For any finite set A ⊂ Q with | AA | ≤ K | A | , any non-zero u ∈ Q and any positiveinteger k , | ( A + u ) ( k ) | ≥ | A | k (8 k ) kK . The proof of this result was based on an argument that Chang [6] introduced to give similarbounds for the k -fold sum set of a set with small product set. Theorem 1.2 is essentially optimalwhen K is of the order c log | A | , for a sufficiently small constant c = c ( k ). However, the resultbecomes trivial when K is larger, for example if K = | A | ǫ and ε >
0. The bulk of this paper isdevoted to proving the following theorem, which gives a near optimal bound for the size of ( A + u ) ( k ) when K = | A | ε , for a sufficiently small but positive ε . Theorem 1.3.
Given < γ < / , there exists a positive constant C = C ( γ, k ) such that for anyfinite A ⊂ Q with | AA | = K | A | and any non-zero rational u , | ( A + u ) ( k ) | ≥ | A | k (1 − γ ) − K Ck . In fact, we prove a more general version of Theorem 1.3 in terms of certain weighted energies andso-called Λ-constants (see Theorem 3.7 for the general statement that implies Theorem 1.3 - seesections 2 and 3 for the relevant definitions of energy and Λ-constants). This more general resultis what allows us to deduce Theorem 1.1.1.2.
A subspace type theorem – an l ∞ sum-product estimate. It appears that Theorem1.1, as well as the forthcoming generalised form of Theorem 1.3, lead to some interesting newapplications. To illustrate the strength of these sum-product results, we present three applicationsin this paper.
N ITERATED PRODUCT SETS WITH SHIFTS II 3
Our main application concerns a variant of the celebrated Subspace Theorem by Evertse, Schmidtand Schlikewei [8] which, after quantitative improvements by Amoroso and Viada [1], reads asfollows.Suppose a , . . . , a k ∈ C ∗ , α , . . . , α r ∈ C ∗ and defineΓ = { α z · · · α z r r , z i ∈ Z } , so Γ is a free multiplicative group of rank r . Consider the equation a x + a x + · · · + a k x k = 1 (2)with a i ∈ C ∗ viewed as fixed coefficients and x i ∈ Γ as variables. A solution ( x , . . . , x k ) to (2) iscalled nondegenerate if for any non-empty J ( { , . . . , k } X i ∈ J a i x i = 0 . Theorem 1.4 (The Subspace Theorem, [8] [1] ) . The number A ( k, r ) of nondegenerate solutionsto (2) satisfies the bound A ( k, r ) ≤ (8 k ) k ( k + kr +1) . (3)The Subspace Theorem dovetails nicely to the following version of the Freiman Lemma. Theorem 1.5.
Let ( G, · ) be a torsion-free abelian group and A ⊂ G with | AA | < K | A | . Then A iscontained in a subgroup G ′ < G of rank at most K . Now assume for simplicity that A ⊂ Q and | AA | ≤ K | A | . Let us call such sets (this definitiongeneralizes of course to an arbitrary ambient group) K - almost subgroups .We now show that it is natural to expect that the Subspace Theorem generalises to K -almostsubgroups with K taken as a proxy for the group rank. A straightforward corollary of Theorem 1.5and Theorem 1.4 is as follows. Corollary 1.6 (Subspace Theorem for K -almost subgroups) . Let A be a K -almost subgroup. Thenthe number A ( k, K ) of non-degenerate solutions ( x , x , . . . , x k ) ∈ A k to c x + c x + . . . c k x k = 1 with fixed coefficients c i ∈ C ∗ is bounded by A ( k, K ) ≤ (8 k ) k ( k + kK +1) . Similarly to Theorem 1, the bound of Corollary 1.6 becomes trivial when A is large and K islarger than c log | A | for some small c > The original theorem is formulated in a more general setting, namely for the division group of Γ, but we will stickto the current formulation for simplicity. One could’ve used a more general framework of K - approximate subgroups introduced by Tao. We decided tointroduce a simpler definition in order to avoid technicalities. However, in the abelian setting the definitions areessentially equivalent. B. HANSON, O. ROCHE-NEWTON, AND D. ZHELEZOV
Conjecture 1.
There is a constant c ( k ) such that Corollary 1.6 holds with the bound A ( k, K ) ≤ K c ( k ) . We can support Conjecture 1 with a special case k = 2 and A ⊂ Q , c i ∈ Q and a somewhatweaker estimate, which we see as a proxy for the Beukers-Schlikewei Theorem [3]. Theorem 1.7 (Weak Beukers-Schlikewei for K -almost subgroups) . For any γ > there is C ( γ ) > such that for any K -almost subgroup A ⊂ Q and fixed non-zero c , c ∈ Q the number A (2 , K ) ofsolutions ( x , x ) ∈ A to c x + c x = 1 is bounded by A (2 , K ) ≤ | A | γ K C . One can view Theorem 1.7 as an l ∞ version of the weak Erd˝os-Szemer´edi sum-product conjecture.The weak Erd˝os-Szemer´edi conjecture is the statement that, if | AA | ≤ K | A | then | A + A | ≥ K − C | A | for some positive absolute constant C . For A ⊂ Z , this result was proved in [4], but the conjectureremains open over the reals.A common approach to proving sum-product estimates is to attempt to show that, for a set A with small product set, the additive energy of A , which is defined as the quantity E + ( A ) := |{ ( a, b, c, d ) ∈ A : a + b = c + d }| , is small. Indeed, this was the strategy implemented in [6] and [4], the latter of which showed that,for all γ >
0, there is a constant C = C ( γ ) such that for any A ⊂ Q with | AA | ≤ K | A | , E + ( A ) ≤ K C | A | γ . (4)Since there are at least | A | trivial solutions when { a, b } = { c, d } , this bound is close to bestpossible. It then follows from a standard application of the Cauchy-Schwarz inequality that | A + A | ≥ | A | − γ K C . Defining the representation function r A + A ( c ) = |{ ( a , a ) ∈ A × A : a + a = c }| , it follows that E + ( A ) = X x r A + A ( x ) , and so bounds for the additive energy can be viewed as l estimates for this representation function.Theorem 1.7 gives the stronger l ∞ estimate: it says that, if | AA | ≤ K | A | then r A + A ( c ) ≤ K C | A | γ for all c = 0. This implies (4), and thus in turn the weak Erd˝os-Szemer´edi sum-product conjecture.We prove Theorem 1.7 in Section 4. This is something of an over-simplification, as [4] in fact proved a much more general result which bounded themulti-fold additive energy with weights attached.
N ITERATED PRODUCT SETS WITH SHIFTS II 5
Remark.
It is highly probable that our method can be combined with the ideas of [5] which wouldgeneralize Theorem 1.7 to K -almost subgroups consisting of algebraic numbers of degree at most d (though not necessarily contained in the same field extension). The upper power C is going todepend on d then, so the putative bound (using the notation of Theorem 1.7) is A (2 , K ) ≤ C ′ ( d ) | A | γ K C ( γ,d ) with some C, C ′ > . We are going to consider this matter in detail elsewhere. Note, however, thatproving a similar statement with no dependence on d seems to be a significantly harder problem. Further applications.
An inverse Szemer´edi-Trotter Theorem.
Theorem 1.7 can be interpreted as a partial inverseto the Szemer´edi-Trotter Theorem. The Szemer´edi-Trotter Theorem states that, if P is a finite setof points and L is a finite set of lines in R , then the number of incidences I ( P, L ) between P and L satisfies the bound I ( P, L ) := |{ ( p, l ) ∈ P × L : p ∈ l }| = O ( | P | / | L | / + | P | + | L | ) . (5)The term | P | / | L | / above is dominant unless the sizes of P and L are rather imbalanced. TheSzemer´edi-Trotter Theorem is tight, up to the multiplicative constant.It is natural to consider the inverse question: for what sets P and L is it possible that I ( P, L ) =Ω( | P | / | L | / )? The known constructions of point sets which attain many incidences appear toall have some kind of lattice like structure. This perhaps suggests the loose conjecture that pointsets attaining many incidences must always have some kind of additive structure, although such aconjecture seems to be far out of reach to the known methods.However, with an additional restriction that P = A × A with A ⊂ Q , Theorem 1.1 leads to thefollowing partial inverse theorem, which states that if A has small product set then I ( P, L ) cannotbe maximal.
Theorem 1.8.
For all γ ≥ there exists a constant C = C ( γ ) such that the following holds. Let A be a finite set of rationals such that | AA | ≤ K | A | and let P = A × A . Then, for any finite set L of lines in the plane, I ( P, L ) ≤ | P | + | A | γ K C | L | . In fact, not only does this show that I ( A × A, L ) cannot be maximal when | AA | is small, but betterstill the number of incidences is almost bounded by the trivial linear terms in (5). The insistencethat the point set is a direct product is rather restrictive. However, since many applications of theSzemer´edi-Trotter Theorem make use of direct products, it seems likely that Theorem 1.8 could beuseful. The proof is given in Section 10.1.3.2. Improved bound for the size of an additive basis of a set with small product set.
Theorem 1.7also yields the following application concerning the problem of bounding the size of an additivebasis considered in [15]. We can significantly improve the bound in the rational setting, pushingthe exponent in (6) from 1 / / − o ǫ (1) to 2 / − o ǫ (1) in the limiting case K = | A | ǫ . B. HANSON, O. ROCHE-NEWTON, AND D. ZHELEZOV
Theorem 1.9.
For any γ > there exists C ( γ ) such that for an arbitrary A ⊂ Q with | AA | = K | A | and B, B ′ ⊂ Q , S := (cid:12)(cid:12) { ( b, b ′ ) ∈ B × B ′ : b + b ′ ∈ A } (cid:12)(cid:12) ≤ | A | γ K C min {| B | / | B ′ | + | B | , | B ′ | / | B | + | B ′ |} . In particular, for any γ > there exists C ( γ ) such that if A ⊂ B + B then | B | ≥ | A | / − γ K − C . (6)The proof of Theorem 1.9 is given in Section 10. Remark.
During the preparation of the manuscript we became aware that Cosmin Pohoata hasindependently proved Theorem 1.9 using an earlier result of Chang and by a somewhat differentmethod.
Unlimited growth for products of difference sets.
It was conjectured in [2] that for any b ∈ R there exists k = k ( b ) ∈ N such that for all A ⊂ R | ( A − A ) k | ≥ | A | b . In another application of Theorem 1.1, we give a positive answer to this question under the addi-tional restriction that A ⊂ Q . In fact, we prove the following stronger statement. Theorem 1.10.
For any b ∈ R there exists k = k ( b ) ∈ N such that for all A ⊂ Q and B ⊂ Q with | B | ≥ , | ( A + B ) k | ≥ | A | b . The proof is given in Section 10.1.4.
The structure of the rest of this paper.
In section 2, we introduce a new kind of mixedenergy, and establish some initial bounds on this energy which are strong when the multiplicativedoubling K is of the order c log | A | for a sufficiently small constant c . The structure of thesearguments are similar to those introduced by Chang in [6], and also used by the authors in [10].We also introduce the notion of separating constants in section 2, which generalises that of theaforementioned mixed energy.Section 3 begins by stating the crucial Theorem 3.1, which states that is | AA | is small then thereis a large subset A ′ ⊂ A with a good separating constant. The rest of the section introduces thelanguage of Λ-constants and some of their crucial properties. These properties are then used insection 4 to conclude the proofs of the main results of this paper, Theorems 1.1, 1.3 and 1.7, usingTheorem 3.1 as a black box.It then remains to prove Theorem 3.1. This is a long and technical proof, where we need toamplify the bounds obtained in section 2 in several stages. This process happens in sections 5, 6, 7, N ITERATED PRODUCT SETS WITH SHIFTS II 7 Finally, in section 10, we give proofs of furtherapplications of our main results.2.
A Chang-type bound for the mixed energy
Different kinds of energies play a pivotal role in the work of Chang [6] and Bourgain-Chang [4],as well as [10]. In [6], it was proved that, for any finite set of rationals A with | AA | ≤ K | A | , the k-fold additive energy , which is defined as the number of solutions to a + · · · + a k = a k +1 + · · · a k , ( a , . . . , a k ) ∈ A k , (7)is at most (2 k − k ) kK | A | k . A simple application of the Cauchy-Schwarz inequality then impliesthat the k -fold sum set satisfies the bound | kA | ≥ | A | k (2 k − k ) kK . Bound (7) is close to optimal when K = c log | A | , but becomes trivial when K = | A | ε . In [4], (aweighted version of) this bound was used as a foundation, and developed considerably courtesyof some intricate decoupling arguments, in order to prove a bound for the k -fold additive energywhich remains very strong when K is of the order | A | ε .In [10], we followed a similarly strategy to that of [6], proving that for any finite set of rationals A with | AA | ≤ K | A | and any non-zero rational u , the k-fold multiplicative energy of A + u , whichis defined as the number of solutions to( a + u ) · · · ( a k + u ) = ( a k +1 + u ) · · · ( a k + u ) , ( a , . . . , a k ) ∈ A k , (8)is at most ( Ck ) kK | A | k . Unfortunately, in adapting the approach of [6] in order to bound thenumber of solutions to (8) in [10], we encountered some difficulties with dilation invariance whichmade the argument rather more complicated, and we were unable to marry our methods with thoseof [4] to obtain a strong bound when K is of order | A | ε .In this paper, we modify the approach of [10] by working with a different form of energy. Considerthe following representation function: r k ( x, y ) = |{ ( a , . . . , a k ) ∈ A k : a · · · a k = x, ( a + u ) · · · ( a k + u ) = y }| . Then, because r k is supported on A ( k ) × ( A + u ) ( k ) , it follows from the Cauchy-Schwarz inequalitythat | A | k = X ( x,y ) ∈ A ( k ) × ( A + u ) ( k ) r k ( x, y ) ≤ | A ( k ) || ( A + u ) ( k ) | X ( x,y ) ∈ A ( k ) × ( A + u ) ( k ) r k ( x, y ) . (9)The innermost sum is the quantity˜ E k ( A ; u ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ( a , . . . , a k , b , . . . , b k ) ∈ A k : k Y i =1 a i = k Y i =1 b i , k Y i =1 ( a i + u ) = k Y i =1 ( b i + u ) )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We recommend that the reader consult [18] for more information about the proof of the Bourgain-Chang Theorem,and particularly the early parts of [18], where an attempt is made to outline some heuristics of the proof.
B. HANSON, O. ROCHE-NEWTON, AND D. ZHELEZOV
We summarise this in the following lemma.
Lemma 2.1.
For any finite set A ⊂ R , any u ∈ R \ { } and any integer k ≥ , we have | A | k ≤ | A ( k ) || ( A + u ) ( k ) | ˜ E k ( A ; u ) . In particular, | A | k ˜ E k ( A ; u ) / ≤ max {| A ( k ) | , | ( A + u ) ( k ) |} . Our goal is to estimate this energy and to show that, at least for sets of rationals, it cannot everbe too big.In this section we seek to give an initial upper bound for ˜ E k ( A ; u ). The strategy is close to thatof Chang [6]. There are also clear similarities with the prequel to this paper [10].To do this, as in [10], we will write ˜ E k ( A ; u ) in terms of Dirichlet polynomials. In this case, ourDirichlet polynomials will be functions of the form F ( s , s ) = X ( a,b ) ∈ Q f ( a, b ) a s b s where f : Q → C is some function of finite support. It will also be more convenient to countweighted energy. For w a a sequence of non-negative weights on A , let˜ E k,w ( A ; u ) = X a ··· a k = b ··· b k ( a + u ) ··· ( a k + u )=( b + u ) ··· ( b k + u ) w a · · · w a k w b · · · w b k Lemma 2.2.
Let A be a finite set of rational numbers and let u be a non-zero rational number.Then, for any integer k ≥ , we have ˜ E k,w ( A ; u ) = lim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X a ∈ A w a a it ( a + u ) it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt . Proof.
Expanding, the double integral on the right hand side is equal to X a ,...,a k ∈ A X b ,...,b k ∈ A w a · · · w a k w b · · · w b k ·· Z T ( a · · · a k b − · · · b − k ) it dt Z T (( a + u ) · · · ( a k + u )( b + u ) − · · · ( b k + u ) − ) it dt . Now 1 T Z T ( u/v ) it dt = ( u = v,O u,v ( T − ) if u = v. From this, the lemma follows. (cid:3)
Let k · k k be the standard norm in L k [0 , T ] , normalised such that k k k = 1. So, k f k k := (cid:18) T Z T Z T | f ( t ) | k dt (cid:19) / k . N ITERATED PRODUCT SETS WITH SHIFTS II 9
Lemma 2.3.
Let J be a set of integers and decompose it as J = J ∪ · · · ∪ J N . For each j ∈ J let f j : R × R → C be a function belonging to L k (cid:0) R (cid:1) for every integer k ≥ . Then, for everyinteger k ≥ , lim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈J f j ( t , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k ≤ N N X n =1 lim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈J n f j ( t , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k . (10) Proof.
It suffices to prove the inequality for all sufficiently large T , which we assume fixed for now.Then T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈J f j ( t , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X n =1 X j ∈J n f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k ≤ N X n =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ∈J n f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k , (11)by the triangle inequality. By the Cauchy-Schwarz inequality, (11) is bounded by N N X n =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ∈J n f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k . (12)Letting T → ∞ we get the claim of the lemma. (cid:3) Corollary 2.4.
Let A be a finite set of rational numbers, partitioned as A = A ∪ · · · ∪ A N , let w be a set of non-negative weights, and let u be a non-zero rational number. Then for any integer k ≥ E k,w ( A ; u ) /k ≤ N N X j =1 ˜ E k,w ( A j ; u ) /k . Now let p be a fixed prime. For a ∈ Q , let v p ( a ) denote the p -adic valuation of a . For a set A ofrational numbers and an integer t , we let A t = { a ∈ A : v p ( a ) = t } . Lemma 2.5.
Let p be a prime number. Suppose A is a finite set of rational numbers and let u bea non-zero rational number. Then for any w , a set of non-negative weights on A , and any integer k ≥ , ˜ E k,w ( A ; u ) /k ≤ (cid:18) k (cid:19) X d ∈ Z ˜ E k,w ( A d ; u ) /k . Proof.
First, let A = A + ∪ A − where A + = { a ∈ A : v p ( a ) ≥ v p ( u ) } and A − = { a ∈ A : v p ( a ) 0, expanding out both sides of (17) and simplifying gives u − ( a + · · · + a k ) + higher terms = u − ( b k +1 + · · · + b k ) + higher terms . (18)If all of the d i are distinct, then there is some unique smallest d i , and thus a unique smallest valueof v p ( a i ). But then the left hand side and the right hand side are divisible by distinct powers of p ,a contradiction.So returning to (14), we need only consider the cases in which one or more of the d i are repeated.There are three kinds of ways in which this can happen.(1) d i = d ′ i with 1 ≤ i ≤ k and k + 1 ≤ i ′ ≤ k . There are k possible positions for such a pair( i, i ′ ),(2) d i = d ′ i with 1 ≤ i, i ′ ≤ k . There are (cid:0) k (cid:1) possible positions for such a pair ( i, i ′ ),(3) d i = d ′ i with k + 1 ≤ i, i ′ ≤ k . There are (cid:0) k (cid:1) possible positions for such a pair ( i, i ′ ). N ITERATED PRODUCT SETS WITH SHIFTS II 11 Suppose we are in situation (1) above. Specifically, suppose that d = d k . The other k − X d ≥ v p ( u ) lim T →∞ T Z T Z T f d ( t , t ) f d ( t , t ) X d ,...,d k − ≥ v p ( u ) f d ( t , t ) · · · f d k ( t , t ) f d k +1 ( t , t ) · · · f d k − ( t , t ) dt dt = X d ≥ v p ( u ) lim T →∞ T Z T Z T | f d ( t , t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d ≥ v p ( u ) f d ( t , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k − dt dt . (19)Suppose we are in situation (2). Specifically, suppose that d = d . The other (cid:0) k (cid:1) − X d ≥ v p ( u ) lim T →∞ T Z T Z T f d ( t , t ) X d ,...,d k ≥ v p ( u ) f d ( t , t ) · · · f d k ( t , t ) f d k +1 ( t , t ) · · · f d k ( t , t ) dt dt ≤ X d ≥ v p ( u ) lim T →∞ T Z T Z T | f d ( t , t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d ≥ v p ( u ) f d ( t , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X d f d ( t , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt = X d ≥ v p ( u ) lim T →∞ T Z T Z T | f d ( t , t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d ≥ v p ( u ) f d ( t , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k − dt dt . The same argument also works in case (3). Returning to (14), we then have˜ E k,w ( A + ; u ) ≤ (cid:18) k (cid:19) X d ≥ v p ( u ) lim T →∞ T Z T Z T | f d ( t , t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d ≥ v p ( u ) f d ( t , t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k − dt dt ≤ (cid:18) k (cid:19) X d ≥ v p ( u ) ˜ E k,w ( A d ; u ) /k E k,w ( A + ; u ) − /k , the last inequality being H¨older’s. It therefore follows that˜ E k,w ( A + ; u ) /k ≤ (cid:18) k (cid:19) X d ≥ v p ( u ) ˜ E k,w ( A d ; u ) /k . (20)Now we proceed to E k,w ( A − ; u ) /k . For any solution to the pair of equations a · · · a k = a k +1 · · · a k ( a + u ) · · · ( a k + u ) = ( a k +1 + u ) · · · ( a k + u )we have a solution to the equation(1 + ua − ) · · · (1 + ua − k ) = (1 + ua − k +1 ) · · · (1 + ua − k ) . Again, we expand and simplify, using this time that v p ( ua − i ) is positive, and get u ( a − + · · · a − k ) + higher terms = u ( a − k +1 + · · · a − k ) + higher terms . As in the previous case , we cannot have a unique smallest v p ( ua − i ). We can therefore repeatthe arguments that gave us (20) in order to deduce that˜ E k,w ( A − ; u ) /k ≤ (cid:18) k (cid:19) X d Lemma 2.6. Let p , . . . , p K be a prime numbers. Suppose A is a finite set of rational numbers andlet u be a non-zero rational number. For a vector d = ( d , . . . , d K ) , define A d = { a ∈ A : v p ( a ) = d , . . . , v p k ( a ) = d k } . Then for any w , a set of non-negative weights on A , and for any integer k ≥ , ˜ E k,w ( A ; u ) /k ≤ (cid:18) (cid:18) k (cid:19)(cid:19) K X d ∈ Z K ˜ E k,w ( A d ; u ) /k . Proof. The aim is to prove thatlim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d ∈ Z K X a ∈ A d w a a it ( a + u ) it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k ≤ (cid:18) (cid:18) k (cid:19)(cid:19) K X d ∈ Z K lim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ A d w a a it ( a + u ) it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k . (22) Note that here we have used the information that a · · · a k = a k +1 · · · a k , whereas we did not use this whenbounding ˜ E k,w ( A + ; u ). N ITERATED PRODUCT SETS WITH SHIFTS II 13 We proceed by induction on K , the base case K = 1 being given by Lemma 2.5. Thenlim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d ∈ Z K X a ∈ A d w a a it ( a + u ) it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k = lim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d K ∈ Z X d ′ ∈ Z K − X a ∈ A ( d ′ ,d ) w a a it ( a + u ) it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k ≤ (cid:18) k (cid:19) X d K ∈ Z lim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d ′ ∈ Z K − X a ∈ A ( d ′ ,d ) w a a it ( a + u ) it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k ≤ (cid:18) k (cid:19) X d K ∈ Z (cid:18) (cid:18) k (cid:19)(cid:19) K − X d ′ ∈ Z K − lim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ A ( d ′ ,d ) w a a it ( a + u ) it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k = (cid:18) (cid:18) k (cid:19)(cid:19) K X d ∈ Z K lim T →∞ T Z T Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ A d w a a it ( a + u ) it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt dt /k . The first inequality above follows from an application of Lemma 2.5. The second inequality followsfrom the induction hypothesis. (cid:3) Separating constants. The following definition, which follows the terminology used in [18],is central to this paper. Let ψ be an arbitrary real number. A set X ⊂ Q is said to be ψ -separating if for any non-zero u ∈ Q , any set finite Z ⊂ Q of the form Z = [ x ∈ X xY x such that ( x, Y x ′ ) = 1 for all x, x ′ ∈ X , and any set of weights w on Z ˜ E k,w ( Z ; u ) /k ≤ ψ X x ∈ X ˜ E k,w ( xY x ; u ) /k . A first observation about separating constants comes in the form of the following claim. Claim 2.7. Any finite A ⊂ Q is | A | -separating. This claim follows immediately from Corollary 2.4. Combining this new definition with Lemma2.5, we can also record the following corollary. Corollary 2.8. Let p be a prime number. Suppose A is of the form A = { p h : h ∈ H } for somefinite set H ⊂ Z . Then A is (cid:0) k (cid:1) -separating. Strictly speaking, we should perhaps include k in this definition and say that a set is ( ψ, k )-separating if itsatisfied the stated conditions. In order to simplify the notation we do not do this. Instead, we can think of k ≥ k willbe omitted from statements of results. With this definition of the separating constant, we can use Lemma 2.6 to get a first bound forthe separating constant of a set with small product set. Once again, this bound is good when K ≤ c log | A | for a sufficiently small constant c .To do this, we recall an argument of Chang [6] which uses Freiman’s Lemma to show that a setof rationals with small product set is determined by a small number of prime factors. Let A be aset of rationals and let P := { p : p is prime and there exists a ∈ A, v p ( a ) = 0 } = { p , . . . , p t } be the set of primes dividing some element of A . Abusing notation slightly, we define a map P : A → Z t where P ( a ) = ( v p ( a ) , . . . , v p t ( a )). Denoting by P ( X ) the image of a set X under P ,observe that P ( AA ) = P ( A ) + P ( A ). We define the multiplicative dimension of A ⊂ Q to be theleast dimension of an affine space L containing P ( A ). Theorem 2.9 (Freiman’s Lemma) . Let A ⊂ R m be a finite set not contained in a proper affinesubspace. Then | A + A | ≥ ( m + 1) | A | − O m (1) . Theorem 2.10. Let A ⊂ Q be finite with | AA | = K | A | . Then, A is (cid:16) (cid:0) k (cid:1)(cid:17) K -separating.Proof. It follows from Freiman’s Lemma that if | AA | ≤ K | A | with | A | sufficiently large, then A hasmultiplicative dimension at most K .This means that there is a set of { p , . . . , p K } of primes and a set of vectors J ⊂ Z K such that A = [ j =( j ,...,j K ) ∈J p j · · · p j K K x j , where each x j is a rational number coprime to p · · · p K . For j = ( j , . . . , j K ) ∈ J , write a j = p j · · · p j K K x j .Now, let Z = [ j ∈J a j Y j ⊂ Q with the ( Y j , a j ′ ) = 1 for all j , j ′ ∈ J . In particular, Y j is coprime to p · · · p K . Therefore, in thenotation of Lemma 2.6 Z j = a j Y j . Then, by Lemma 2.6, ˜ E k,w ( Z ; u ) /k ≤ (cid:18) (cid:18) k (cid:19)(cid:19) K X j ∈ Z K ˜ E k,w ( a j Y j ; u ) /k . (cid:3) We recall now the Pl¨unnecke-Ruzsa Theorem. See [14] for a simple inductive proof. Followingconvention, we state it using additive notation, although it will be used in the multiplicative setting. We say that two rational numbers a and b are comprime if at least one of v p ( a ) and v p ( b ) is zero for all prime p .As with the case of integers, we write ( a, b ) = 1. N ITERATED PRODUCT SETS WITH SHIFTS II 15 Theorem 2.11. Let A be a subset of a commutative additive group G with | A + A | ≤ K | A | . Thenfor any h ∈ N , | hA | ≤ K h | A | . One may think of the separating constant of X as a generalisation of the notion of the mixedenergy ˜ E k ( X ; u ). Indeed, if X is ψ -separating then take Y x = { } for all x ∈ X and w ( x ) = 1for all x ∈ X . Then it follows that ˜ E k ( X ; u ) ≤ ψ k | X | k . In particular, Theorem 2.10 implies thefollowing result. Theorem 2.12. Let A be a finite set of rational numbers and let u ∈ Q be non-zero. Suppose that | AA | ≤ K | A | . Then, for any integer k ≥ , | ( A + u ) k | ≥ | A | k − K k (cid:16) (cid:0) k (cid:1)(cid:17) Kk Proof. By Theorem 2.10, ˜ E k ( A ; u ) ≤ (cid:18) (cid:18) k (cid:19)(cid:19) Kk | A | k . Also, by Theorem 2.11, | A ( k ) | ≤ K k | A | . Inserting these two bounds into Lemma 2.1 completes theproof. (cid:3) A stronger version of Theorem 2.12 was the main result of [10], which used the standard k -foldmultiplicative energy of the set A + u .A key goal of this paper is to amplify this approach in order to give a good bound for the casewhen K = | A | ε . The advantage of working with this generalised notion of energy is that it has acrucial “chaining property” which will be important in the forthcoming analysis for pushing to getresults for larger K . Lemma 2.13. Let A be a finite set of rationals which can be decomposed as a disjoint union A = [ b ∈ B bC b and with ( b, C b ′ ) = 1 for all b, b ′ ∈ B . Assume also that B is ψ -separating and that each C b is ψ -separating. Then A is ψ ψ -separating.Proof. Let Z be a set of rationals which decomposes as Z = [ a ∈ A aY a with ( a, Y a ′ ) = 1 for all a, a ′ ∈ A . Then for any u ∈ Q and weights w on Z ,˜ E w,u ( Z ; u ) /k = ˜ E w,u [ b ∈ B b [ c ∈ C b cY bc ; u /k ≤ ψ X b ∈ B ˜ E w,u b [ c ∈ C b cY bc ; u /k . In the inequality above we have used the fact that B is ψ separating and that b, [ c ∈ C b ′ cY b ′ c = 1for any b, b ′ ∈ B . Indeed, take an arbitrary product cy with c ∈ C b ′ and y ∈ Y b ′ c . Then b is coprimeto c by the hypothesis of the lemma. Also, y is coprime to each element of A by the definition of Z , which implies that y is coprime to b by the hypothesis of the lemma.We therefore have˜ E w,u ( Z ; u ) /k ≤ ψ X b ∈ B ˜ E w,u [ c ∈ C b c ( bY bc ); u /k ≤ ψ X b ∈ B ψ X c ∈ C b ˜ E w,u ( c ( bY bc ); u ) /k = ψ ψ X a ∈ A ˜ E w,u ( aY a ; u ) /k . The inequality above uses fact that C b is ψ -separating and that ( c, bY bc ′ ) = 1 for any c, c ′ ∈ C b .This can be verified in the same way as the previous inequality. (cid:3) Lambda-constants We will soon begin the process of amplifying Theorem 2.10 from the previous section in orderto get a better separating factor which leads to strong bound when K = | A | ε . At the conclusion ofthis process we will prove the following result. Theorem 3.1. Given < τ, γ < / , there exist positive constants C = C ( τ, γ, k ) and C = C ( τ, γ, k ) such that for any finite A ⊂ Q with | AA | = K | A | , there exists A ′ ⊂ A with | A ′ | ≥ K − C | A | − τ such that A ′ is K C | A | γ -separating. In fact, one can check that the proof of Theorem 3.1 goes through in a more general setting. Let S : 2 Q → R be a function defined on rational sets with the following properties:(1) (Trivial bound) For an arbitary set A ⊂ Q S ( A ) ≤ | A | . (2) (Stability) If A ′ ⊂ A then S ( A ′ ) ≤ S ( A ) . (3) ( p -adic separation) There is an absolute constant s ≥ p and I ⊂ Z , S ( [ i ∈ I p i ) ≤ s. (4) (Nesting) Let A ⊂ Q and { B a } a ∈ A is a collection of sets such that ( a, B a ′ ) = 1 for any a, a ′ ∈ A . Further assume that aB a , a ∈ A are pairwise disjoint. Then S ( [ a ∈ A aB a ) ≤ S ( A ) max a ∈ A S ( aB a ) . N ITERATED PRODUCT SETS WITH SHIFTS II 17 Note that our definition of the separating constant satisfies (1)-(4). Theorem 3.2. Let S be a function with the properties above. Given < τ, γ < / , there existpositive constants C = C ( τ, γ, s ) and C = C ( τ, γ, s ) such that for any finite A ⊂ Q with | AA | = K | A | , there exists A ′ ⊂ A with | A ′ | ≥ K − C | A | − τ such that S ( A ′ ) ≤ K C | A | γ . The proof of Theorem 3.1 is essentially borrowed from [4]. We present here a proof adapted toour setting to make the paper self-contained. The same proof applies to Theorem 3.2 with cosmeticmodifications, but we expect that it might be useful to have such a general ‘black-box’ version forfuture use.Before we begin the lengthy proof of Theorem 3.1, we will take some time to see how it impliesthe two main theorems of this paper. To do this, it will be convenient to use the language of Λ -constants , and to introduce some of their key properties. The main motivation behind Λ-constantsis the stability property given by the forthcoming Corollary 3.4, which is absent in the non-weightedversion of the energy.We also encourage the interested reader to consult our preceding paper [10] for a slightly moregentle introduction to Λ-constants in the setting of Dirichlet polynomials and more in-depth moti-vation behind this concept.Let A ⊂ Q be a finite set and let u be a non-zero rational. DefineΛ k ( A ; u ) := max ˜ E k,w ( A ; u ) /k , where the maximum is taken over all weights w on A such that X a ∈ A w ( a ) = 1 . (23)An equivalent definition isΛ k ( A ; u ) := max lim T →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X a ∈ A w a a it ( a + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k . where the maximum is taken over the same range of weights. Lemma 3.3. Let A ⊂ Q be a finite set with some non-negative real weights w a assigned to eachelement a ∈ A and let u be a non-zero rational. Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X a ∈ A w a a it ( a + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k ≤ Λ k ( A ; u ) X a ∈ A w a ! + o T →∞ (1) . (24) Proof. If P a ∈ A w a = 0 the claim of the lemma is trivial. Otherwise, define new weights w ′ a := w a ( P a ∈ A w a ) / 28 B. HANSON, O. ROCHE-NEWTON, AND D. ZHELEZOV which satisfy (23). It thus suffices to show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X a ∈ A w ′ a a it ( a + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k ≤ Λ k ( A ; u ) + o T →∞ (1) , which is a straightforward consequence of our definition of Λ k ( A ; u ). (cid:3) We will use the following stability property of Λ-constants which helps us to work with subsets. Corollary 3.4. Suppose that A ⊂ Q , that u is a non-zero rational and A ′ ⊂ A . Then Λ k ( A ′ ; u ) ≤ Λ k ( A ; u ) . In particular, ˜ E /kk ( A ′ ; u ) ≤ Λ k ( A ; u ) | A ′ | . and ˜ E k ( A ; u ) ≤ Λ kk ( A ; u ) | A | k . Proof. The first claim follows from the observation that any set of weights { w a } a ∈ A ′ with P w a = 1can be trivially extended to a set of weights { w a } a ∈ A by assigning zero weight to the elements in A \ A ′ . Next observe that E k is just E k,w with all the weights being one and apply Lemma 3.3. (cid:3) The next lemma records that any set with small separating factors also has a small Λ-constant. Lemma 3.5. Let A ⊂ Q be ψ -separating. Then for any u ∈ Q \ { } Λ k ( A ; u ) ≤ ψ. Proof. Let w be any set of weights on A that satisfy (23). Write A = [ a ∈ A aY a with Y a = { } for all a ∈ A . Then by the definition of ψ -separating, it follows that for any non-zero u ∈ Q ˜ E k,w ( A ; u ) /k ≤ ψ X a ∈ A ˜ E k,w ( { a } ; u ) /k = ψ X a ∈ A ( w ( a ) k ) /k = ψ. (cid:3) Lemma 3.6. Let A ⊂ Q be a finite set with | AA | ≤ K | A | and let u be a non-zero rational number.Suppose that A ′ ⊂ A and A ′ is ψ -separating. Then Λ k ( A ; u ) ≤ K (cid:18) | A || A ′ | − (cid:19) ψ. Proof. Let w be an arbitrary set of weights on A such that P a ∈ A w ( a ) = 1. We seek a suitableupper bound for (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X a ∈ A w a a it ( a + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k . N ITERATED PRODUCT SETS WITH SHIFTS II 19 For a fixed z ∈ A/A ′ , define a set of weights w ( z ) on zA ′ by taking w ( z ) ( za ′ ) = w ( za ′ ) if za ′ ∈ A and w ( z ) ( za ′ ) = 0 otherwise. Define R ( A/A ′ ) ,A ′ ( x ) := |{ ( s, a ) ∈ ( A/A ′ ) × A ′ : sa = x }| and note that R ( A/A ′ ) ,A ′ ( x ) ≥ | A ′ | − x ∈ A . This is because, for all non-zero a ′ ∈ A ′ , x = ( xa ′ ) a ′ . Therefore, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X z ∈ A/A ′ X a ′ ∈ A ′ w ( z ) ( za ′ )( za ′ ) it ( za ′ + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X a ∈ A R ( A/A ′ ) ,A ′ ( a ) w ( a ) a it ( a + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k ≥ | A ′ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X a ∈ A w a a it ( a + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k . On the other hand, by the triangle inequality and Lemma 3.3 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X z ∈ A/A ′ X a ′ ∈ A ′ w ( z ) ( za ′ )( za ′ ) it ( za ′ + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k ≤ X z ∈ A/A ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ′ ∈ A ′ w ( z ) ( za ′ )( za ′ ) it ( za ′ + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k ≤ X z ∈ A/A ′ Λ k ( zA ′ ; u ) / + o T →∞ (1) . Since A ′ is ψ -separating, it follows from Lemma 3.5 that Λ k ( zA ′ ; u ) = Λ k ( A ′ ; u/z ) ≤ ψ . We alsohave | A/A ′ | ≤ | A/A | ≤ | AA | | A | ≤ K | A | , by the Ruzsa Triangle Inequality (see [17]). It therefore follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X a ∈ A w a a it ( a + u ) it (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k ≤ K (cid:18) | A || A ′ | − (cid:19) ψ / + o T →∞ (1) , and the result follows. (cid:3) Combining this with Theorem 3.1 gives the following, which is our main result concerning Λ-constants. Theorem 3.7. Given < γ < / , there exists a positive constants C = C ( γ, k ) such that for anyfinite A ⊂ Q with | AA | = K | A | and any non-zero rational u , Λ k ( A ; u ) ≤ K C | A | γ . Concluding the proofs In this section we conclude the proof of Theorem 1.1, which is the main theorem of this paper, andTheorem 1.7 announced in the introduction. Both theorems are restated below for the convenienceof the reader. Theorem 4.1. For all b ∈ Z , there exists k = k ( b ) such that for any finite set A ⊂ Q and anynon-zero rational u , max {| A ( k ) | , | ( A + u ) ( k ) |} ≥ | A | b Proof. Fix b and assume that | A ( k ) | < | A | b for some sufficiently large k = 2 l . The value of l (and thus also that of k ) will be specified at theend of the proof. Since | A (2 l ) | < | A | b , it follows that | A (2 l ) || A (2 l − ) | | A (2 l − ) || A (2 l − ) | · · · | A (2) || A | < | A | b − and thus there is some integer l ≤ l such that | A (2 l ) || A (2 l ) | < | A | b − l . Therefore, writing k = 2 l and B = A ( k ) , we have | BB | < | B || A | b − l . Also, for any non-zero λ ∈ Q , | ( λB )( λB ) | < | B || A | b − l . Therefore, by Theorem 3.7,Λ h ( λB ; u ) ≤ | A | C b − l | B | γ ≤ | A | C b − l + γb where C = C ( h, γ ) and h, γ will be specified later.Now, for some λ ∈ Q , we have A ⊂ λB , and thus by Corollary 3.4 and Lemma 2.1 | A | max {| A ( h ) | , | ( A + u ) ( h ) |} /h ≤ ˜ E /hh ( A ; u ) ≤ | A | Λ h ( λB ; u ) ≤ | A | C b − l + γb . This rearranges to max {| A ( h ) | , | ( A + u ) ( h ) |} ≥ | A | h (1 − C b − l − γb ) . Choose γ = 1 / b and h = 4 b . Then C = C ( h, γ ) = C ( b ) and we havemax {| A ( h ) | , | ( A + u ) ( h ) |} ≥ | A | h (99 / − C ( b ) b − l ) . Then choose l = ( b − C to getmax {| A ( h ) | , | ( A + u ) ( h ) |} ≥ | A | h = | A | b . Note that the choice of l depends only on b and thus k = 2 C ( b − = k ( b ). In particular, since k > h , we conclude that max {| A ( k ) | , | ( A + u ) ( k ) |} ≥ | A | b , as required. (cid:3) Theorem 3.7 also implies Theorem 1.3. The statement is repeated below for the convenience ofthe reader. Theorem 4.2. Given < γ < / and any integer k ≥ , there exists a positive constant C = C ( γ, k ) such that for any finite A ⊂ Q with | AA | = K | A | and any non-zero rational u , | ( A + u ) ( k ) | ≥ | A | k (1 − γ ) − K Ck . N ITERATED PRODUCT SETS WITH SHIFTS II 21 Proof. Define w ( a ) = 1 / | A | / for all a ∈ A and note that (23) is satisfied. Furthermore, for thisset of weights w , ˜ E k,w ( A ; u ) = ˜ E k ( A ; u ) | A | k ≥ | A | k | A ( k ) || ( A + u ) ( k ) | , (25)where the inequality comes from Lemma 2.1. It follows from Theorem 3.7 that there exists aconstant C = C ( γ, k ) such that for any u ∈ Q \ { } , Λ k ( A ; u ) ≤ K C | A | γ . Consequently, by thedefinition of Λ k ( A ; u ), ˜ E k,w ( A ; u ) ≤ K Ck | A | γk . Combining this with (25), it follows that | A ( k ) || ( A + u ) ( k ) | ≥ | A | k (1 − γ ) K Ck . (26)Finally, since | AA | ≤ K | A | , it follows from the Pl¨unnecke-Ruzsa Theorem that | A ( k ) | ≤ K k | A | .Inserting this into (26) completes the proof. (cid:3) We now turn to the proof of Theorem 1.7. Recall its statement. Theorem 4.3. For any γ > there is C ( γ ) > such that for any K -almost subgroup A ⊂ Q andfixed non-zero c , c ∈ Q the number A (2 , K ) of solutions ( x , x ) ∈ A to c x + c x = 1 is bounded by A (2 , K ) ≤ | A | γ K C . Proof. Let S ⊂ A be the set of x ∈ A such that c x + c x = 1 for some x ∈ A . Since theprojection ( x , x ) → x is injective, it suffices to bound the size of S .Since S ⊂ A , by Theorem 3.7 and Corollary 3.4 for any non-zero u ˜ E k ( S ; u ) ≤ K kC ( γ ′ ,k ) | A | kγ ′ | S | k with the parameters 0 < γ ′ < / , k ≥ | S | k ≤ (cid:16) K kC ( γ ′ ,k ) | A | kγ ′ | S | k (cid:17) / max {| S k | , | ( S − /c ) k |} . On the other hand, S ⊆ A and ( S − /c ) ⊆ ( c /c ) A , so by the Pl¨unnecke-Ruzsa inequalitymax {| S k | , | ( S − /c ) k |} ≤ | A ( k ) | ≤ K k | A | . We then have | S | ≤ | A | γ ′ +2 /k K C +2 , and taking k = ⌊ /γ ′ ⌋ + 1 and γ ′ = γ/ 2, the claim follows. (cid:3) Graph Fibering Suppose Z and Z abelian groups, with finite subsets A, B ⊂ Z × Z . We will write z ⊕ z foran element of Z × Z . We will write, for x ∈ X ⊂ Z × Z with π ( x ) = x , X ( x ) = { x ∈ π ( X ) : x ⊕ x ∈ X } . Suppose G ⊂ A × B . Denote by π and π the projections onto the first and second coordinatesof Z × Z respectively. The set G is interpreted as a bipartite graph on A and B , and it can bedecomposed into a union by considering the fibers of π . Indeed, let G = { ( π ( a ) , π ( b )) : ( a, b ) ∈ G } and for ( a , b ) ∈ G , let G ( a , b ) = { ( a , b ) : ( a ⊕ a , b ⊕ b ) ∈ G } ⊂ π ( A ) × π ( B ) . Recall the notation A + G B = { a + b : ( a, b ) ∈ G } . One of the primary reasons for decomposing a graph this way is that it behaves nicely withaddition along the graph. Lemma 5.1. Suppose A and B are finite subsets of Z × Z . Then for G ⊂ A × B we have | A + G B | ≥ | π ( A ) + G π ( B ) | min ( a ,b ) ∈ π ( A ) × π ( B ) | A ( a ) + G ( a ,b ) B ( b ) | . Proof. Write A + G B ⊇ [ s ∈ π ( A + G B ) [ ( a ⊕ a ,b ⊕ b ) ∈ Ga + b = s { ( s ⊕ ( a + b )) } . Next, from the observation that the first union above is disjoint, and the fact that π ( A + G B ) = π ( A ) + G π ( B ), we have | A + G B | ≥ X s ∈ π ( A )+ G π ( B ) (cid:12)(cid:12) [ ( a ⊕ a ,b ⊕ b )) ∈ Ga + b = s { ( s ⊕ ( a + b )) } (cid:12)(cid:12) . Since, for fixed a , b , [ ( a ⊕ a ,b ⊕ b ) ∈ Ga + b = s { a + b } ⊇ A ( a ) + G ( a ,b ) B ( b )the lemma follows. (cid:3) Lemma 5.2 (Regularized decomposition) . Let Z and Z be abelian groups and let A, B ⊂ Z × Z be finite sets. Suppose that δ > , K ≥ and G ⊂ A × B are such that | G | ≥ δ | A || B | , and | A + G B | ≤ K ( | A || B | ) / . N ITERATED PRODUCT SETS WITH SHIFTS II 23 There are absolute constants c, C > , subsets A ′ ⊂ A and B ′ ⊂ B , and a subset G ′ ⊂ A ′ × B ′ withthe following properties.(1) (Uniform fibers) If M A = | π ( A ) | , M B = | π ( B ) | (27) then there are numbers m A and m B satisfying M A m A ≥ cδ (log( K/δ )) − | A | , (28) M B m B ≥ cδ (log( K/δ )) − | B | , (29) m A , m B ≥ cδ K − max a ∈ π ( A ) ,b ∈ π ( B ) ( | A ( a ) | + | B ( b ) | ) , (30) and such that we have approximately uniform fibers: | ( A ′ ) ( a ) | ≈ m A , | ( B ′ ) ( b ) | ≈ m B (31) for a ∈ π ( A ′ ) and b ∈ π ( B ′ ) .(2) (Uniform graph fibering) For some δ , δ > satisfying δ δ > c (log( K/δ )) − δ (32) we have that the first coordinate subgraph is dense: | G ′ | ≥ δ M A M B , (33) and that the subgraph has dense fibers: for each ( a , b ) ∈ G ′ we have | G ′ ( a , b ) | ≥ δ m A m B . (34) (3) (Bounded doubling) For some K , K > with K K ≤ Cδ − (log K ) K (35) we have | π ( A ′ ) + G ′ π ( B ′ ) | = K ( M A M B ) / , (36) and for each ( a , b ) ∈ G ′ , | π ( A ′ ) + G ′ ( a ,b ) π ( B ′ ) | ≈ K ( m A m B ) / . (37)5.1. Proof of Theorem 5.2. We will produce the sets A ′ and B ′ after a sequence of refinements.One such refinement comes from the following lemma. Here, and in what follows, when G ⊆ A × B we write deg G a (respectively, deg G b ) for the size of { b ′ ∈ B : ( a, b ′ ) ∈ G } (respectively, the size of { a ′ ∈ A : ( a ′ , b ) ∈ G } ). Lemma 5.3. Let A and B be finite sets and G ⊆ A × B of size δ | A || B | . Then there exist A ′ ⊂ A, B ′ ⊂ B and G ′ ⊂ G ∩ ( A ′ × B ′ ) such that • deg G ′ a ≥ δ | B | , • deg G ′ b ≥ δ | A | , • | A ′ | ≥ δ | A | , • | B ′ | ≥ δ | B | , and • | G ′ | ≥ δ | A || B | for any a ∈ A ′ , b ∈ B ′ .Proof. Remove from A (respectively, B ) one by one all vertices with degree less than δ | A | / δ | B | / A and B contain only vertices of degree at least δ | A | / δ | B | / 4) in the remaining graph. At the end of this process, we cannot have removed more than δ | A || B | / δ | B | / | A | (and we can remove at most | A | such vertices) or else at most δ | A | / B (and we can remove at most | B | such vertices). Take A ′ and B ′ to be the sets of survivedvertices in A and B respectively and G ′ := G ∩ ( A ′ × B ′ ). (cid:3) Now, set | A | = N A and | B | = N B . In view of the above lemma, and passing to subsets ifnecessary, we may assume | A | ≥ δN A , | B | ≥ δN B , | G | ≥ δN A N B and that for any a ∈ A and b ∈ B we havedeg G a ≥ δ | B | , deg G b ≥ δ | A | . First, we may assume without loss of generality that n A = max a ∈ π ( A ) | A ( a ) | ≥ max b ∈ π ( B ) | B ( b ) | It is also useful to observe that, if a ∈ A then |{ a } + G B | = deg G a , where deg G a is the number ofneighbours of a in G . So, δN B ≤ N A X a ∈ A deg G a ≤ | A + G B | ≤ K ( N A N B ) / . We can apply the same argument, reversing the roles of A and B , and we have proved δN / B ≤ KN / A , δN / A ≤ KN / B . (38)Having assumed this, our first order of business is to establish property (1) for B ′ .5.1.1. Regularization of B . Let a ∈ π ( A ) be such that | A ( a ) | = n A . Then a ⊕ A ( a ) consistsof n A elements of A each with at least δ | B | neighbours in B . Thus | ( a ⊕ A ( a ) × B ) ∩ G | ≥ δn A | B | . Let B ′ = (cid:26) b ∈ B : |{ a : ( a ⊕ a , b ) ∈ G }| ≥ δn A (cid:27) | B ′ | ≥ δ | B | ≥ δ N B (39)and such that for each b ∈ B ′ we have | (( a ⊕ A ( a )) × { b } ) ∩ G | ≥ δn A . N ITERATED PRODUCT SETS WITH SHIFTS II 25 Moreover, since every element in B has at least δN A neighbours in A , we have | ( A × B ′ ) ∩ G | ≥ δN A | B ′ | . If k = | π ( B ′ ) | then there are elements b ⊕ b ′ , . . . , b k ⊕ b ′ k with the b i distinct, and for each of themthe sets a ⊕ A ( a ) + b i ⊕ b ′ i are disjoint, since their first coordinates are a + b i and are distinct. Each of these sets contains atleast δn A distinct elements of A + G B since each element of B ′ has that many neighbours in G .From this it follows that | a ⊕ A ( a ) + G B ′ | ≥ δn A | π ( B ′ ) | and so 18 δn A | π ( B ′ ) | ≤ | A + G B | ≤ K ( N A N B ) / ≤ K δ N B . Here we have used the inequality (38). Next, we define B ′′ = [ ≤ i ≤ k | B ′ ( b i ) |≥ − δ K − n A b i ⊕ B ′ ( b ) . (40)By (40) and (39), | B ′ \ B ′′ | ≤ | π ( B ′ ) | − δ K − n A ≤ − δ N B ≤ δ | B ′ | . Now, we have already assumed that max b ∈ π ( B ) | B ( b ) | ≤ n A , so applying a dyadic partition tothe range 10 − δ K − n A ≤ m ≤ n A , we find a value of m B in this range and a subset B ′′′ = [ b ∈ π ( B ′ ) m B ≤| B ′′ ( b ) |≤ m B b ⊕ B ′ ( b )which has size | B ′′′ | ≫ log( K/δ ) − | B ′′ | . Thus | B ′′′ | ≫ | B ′′ | log( K/δ ) ≫ | B ′ | log( K/δ ) ≫ δ log( K/δ ) N B . Since each element of B has at δN A neighbours in G , we further have | ( A × B ′′′ ) ∩ G | ≥ δN A | B ′′′ | . If M B = | π ( B ′′′ ) | , then because each element of π ( B ′′′ ) has about m B fibers, we have | B ′′′ | ≈ m B M B . Redefine B ′ = B ′′′ and N ′ B = | B ′ | . Then we have shown that N ′ B ≫ δ log( K/δ ) N B . Regularization of A . Let A ′ = [ a ∈ π ( A ) | A ( a ) |≥ − δ K − m B a ⊕ A ( a ) . We first estimate | A \ A ′ | . We write A ′′ = A \ A ′ , so that for each a ∈ A ′′ we have | A ( a ) | < − δ K − m B . (41)We will show | ( A ′′ × B ′ ) ∩ G | ≤ δ N A N ′ B . To see why, assume the contrary. Then there is a b ∈ π ( B ′ ) with | ( A ′′ × b ⊕ B ′ ( b )) ∩ G | ≥ δ N A m B . Indeed, each of the vertex sets b ⊕ B ′ ( b ) are disjoint and have size m B up to a factor of 2. Nowlet A ′′′ ⊂ A ′′ be the set of those a for which | ( { a } × b ⊕ B ′ ( b )) ∩ G | ≥ δ m B . From the definition, it follows that | A ′′′ | ≥ δ N A . (42)Let M = max a ∈ π ( A ′′′ ) | A ′′′ ( a ) | . We have | A ′′′ + G ( b ⊕ B ′ ( b ))) | ≤ | A + G B | ≤ K ( N A N B ) / ≤ K δ N A . Because every element of A ′′′ has at least ( δ/ m B neighbours in b ⊕ B ′ ( b ), and because foreach a ∈ π ( A ′′′ ) the sets ( a ⊕ A ′′′ ( a )) + G ( b ⊕ B ′ ( b )) are disjoint, we get | A ′′′ + G ( b ⊕ B ′ ( b ))) | ≥ ( δ/ m B | π ( A ′′′ ) | . In view of (42) | π ( A ′′′ ) | ≥ | A ′′′ | M ≥ δ M N A , we obtain the bound δ · M N A m B ≤ K δ N A whence M > δ m B K , which contradicts (41) and the definition of M . By what we have just shown, | ( A ′ × B ′ ) ∩ G | ≥ δ N A N ′ B . Now, for each a ∈ A ′ , we certainly have | A ( a ) | ≤ n A ≤ m B δ − K the final estimate coming from the bounds on the range range of m B . Thus we partition the range10 − δ K − m B ≤ | A ( a ) | ≤ m B δ − K 2N ITERATED PRODUCT SETS WITH SHIFTS II 27 dyadically, to find an m A in this range such that A ′′′′ = [ a ∈ π ( A ) m A ≤| A ( a ) |≤ m A a ⊕ A ( a )satisfies | ( A ′′′′ × B ′ ) ∩ G | ≫ δ log( K/δ ) N A N ′ B . Moreover, since | ( A ′′′′ × B ′ ) ∩ G | ≤ | A ′′′′ | N ′ B we have | A ′′′′ | ≫ δ (log( K/δ )) − N A . If we define M A = | π ( A ′′′′ ) | then we have | A ′′′′ | ≈ M A m A as needed. We relabel A ′ = A ′′′′ and N ′ A = | A ′ | , observing that N ′ A ≫ δ log( K/δ ) N A and we are ready to proceed to the next step.5.1.3. Regularizing the graph fibers. So far we have found subsets A ′ and B ′ , and an absoluteconstant c > 0, satisfying | ( A ′ × B ′ ) ∩ G | ≥ c δ log( K/δ ) | A ′ || B ′ | , | A ′ | ≈ m A M A ≥ c δ log( K/δ ) N A , and | B ′ | ≈ m B M B ≥ c δ log( K/δ ) N B . Furthermore, each of A ′ and B ′ have fibers above π of size roughly m A and m B respectively. Recallthat for ( a , b ) ∈ π ( A ′ ) × π ( B ′ ) we have the graph G ( a , b ) = { ( a , b ) ∈ A ′ ( a ) × B ′ ( b ) : ( a ⊕ a , b ⊕ b ) ∈ G } . Because we have regularized the fibers of A ′ and B ′ , each of these graphs has cardinality obeying | G ( a , b ) | ≤ m A m B . By a slight abuse of notation, we let G = { ( π ( a ) , π ( B )) : ( a, b ) ∈ ( A ′ × B ′ ) ∩ G } and define G ′ = (cid:26) ( a , b ) ∈ π ( A ′ ) × π ( B ′ ) : | G ( a , b ) | ≥ cδ 16 log( K/δ ) m A m B (cid:27) . Since X ( a ,b ) ∈ π ( A ′ ) × π ( B ′ ) | G ( a , b ) | = | ( A ′ × B ′ ) ∩ G | ≥ c δ log( K/δ ) | A ′ || B ′ | it follows that X ( a ,b ) ∈ G ′ | G ( a , b ) | ≥ c δ K/δ ) | A ′ || B ′ | . By a dyadic pigeon-holing for δ ′ in the range cδ (log( K/δ )) − ≤ δ ′ ≤ 4, we can find δ ′ ≫ δ (log( K/δ )) − such that G ′′ = (cid:8) ( a , b ) ∈ G ′ : δ ′ m A m B ≤ | G ( a , b ) | ≤ δ ′ m A m B (cid:9) certainly satisfies X ( a ,b ) ∈ G ′′ | G ( a , b ) | ≫ c δ (log( K/δ )) | A ′ || B ′ | . From this estimate, it also follows that | G ′′ | ≫ δδ ′ (log( K/δ )) M A M B . Let us relabel G ′′ as G ′ and set G ′ = { ( a, b ) ∈ A ′ × B ′ : ( π ( a ) , π ( b )) ∈ G ′ } . We move on to the final step of the lemma.5.1.4. Regularizing the doubling constant. For ( a , b ) ∈ π ( A ′ ) × π ( B ′ ) we define K + ( G ( a , b )) = | A ′ ( a ) + G ( a ,b ) B ′ ( b ) | ( | A ′ ( a ) || B ′ ( b ) | ) / . This quantity measure the growth of sumsets on the fibres lying above a pair ( a , b ). Now define H = { ( a , b ) ∈ G ′ : K + ( G ( a , b )) > C (log( K/δ )) δ − K } . Provided C is large enough we have H ≤ | G ′ | . To see this, first observe the trivial bound | π ( A ′ ) + H π ( B ) | ≥ | H | min {| π ( A ′ ) | , | π ( B ′ ) |} ≥ | H | ( M A M B ) / . (43)Let G H = { ( a , a ) ∈ G : ( π ( a ) , π ( a )) ∈ H } ⊂ G. Also, for ( a , b ) ∈ H we have ( G H ) ( a , b ) = G ( a , b )so that by Lemma 5.1 | A ′ + G B ′ | ≥ | π ( A ′ ) + H π ( B ′ ) | min ( a ,b ) ∈ H ( | A ′ ( a ) + G ( a ,b ) B ′ ( b ) | ) . By the definition of H and (43) we see K ( N A N B ) / ≥ | A ′ + G B ′ | ≥ C | H | ( M A M B ) / (log( K/δ )) δ − K ( m A m B ) / . Using our estimates for m A M A , m B M B and G ′ , the right hand side is C | H | M A M B (log( K/δ )) δ − K ( M A m A M B m B ) / ≥ cCK ( N A N B ) / | H || G ′ | . Thus for C sufficiently large in terms of c (which was absolute), we have | H | ≤ | G ′ | . Now let G ′′ = G ′ \ H . We perform yet another dyadic pigeon-holing to find K ′ ≤ C (log( K/δ )) δ − K suchthat G ′′′ = { ( a , b ) ∈ G ′′ : K ′ ≤ K + ( G ( a , b )) ≤ K ′ } N ITERATED PRODUCT SETS WITH SHIFTS II 29 has cardinality | G ′′′ | ≫ | G ′ | log( K/δ ) . Now, by Lemma 5.1 along the subgraph of G with first projection equal to G ′′′ we have K ( N A N B ) / ≥ | π ( A ′ ) + G ′′′ π ( B ′ ) | K ′ ( m A m B ) / = K + ( G ′′′ ) K ′ ( M A m A M B m B ) / , where K + ( G ′′′ ) = | π ( A ′ ) + G ′′′ π ( A ′ ) | ( M A M B ) − / . By the established bounds on m A M A and m B M B , we get K ( N A N B ) / ≫ K + ( G ′′′ ) K ′ δ / log( K/δ )( N A N B ) / . From this we see K + ( G ′′′ ) K ′ ≪ K log( K/δ ) δ / ≪ K log( K ) δ . Now let G ′ = { ( a, b ) ∈ A ′ × B ′ : ( π ( a ) , π ( b )) ∈ G ′′′ } . Define K = K + ( G ′′′ ) and K = K ′ . Let δ = δ ′ and δ = cδ ( δ (log( K/δ )) ) − . One then verifies that with these parameters, the claims ofthe lemma have all been justified. 6. Iteration scheme In this section we will use Lemma 5.2 in order to setup an iteration scheme. At each step wehave a pair of sets ( A , B ) which correspond to a pair of additive sets ( A, B ) := ( P ( A ) , P ( B )) anda graph G on A × B , together with the data ( N, δ, K ) such that:(1) | A || B | = N (2) | A + G B | ≤ KN / (3) | G | ≥ δN .Apart from that, the setup above is equipped with a pair of functions ψ ( N, δ, K ), φ ( N, δ, K )(which are called admissible in [4]). These functions are technical aids to carry out an inductiontype argument. Definition 6.1 (Admissible pair of functions) . A pair of functions ψ ( N, δ, K ) , and φ ( N, δ, K ) issaid to be admissible if for arbitrary sets A, B ⊂ Z [ n ] and a graph G on A × B satisfying (1)-(3)the following holds.There is a graph G ′ ⊆ G such that(G) Graph size is controlled by φ : | G ′ | ≥ φ ( N, δ, K ) (S) Separation of G ′ -neigborhoods is controlled by ψ :For any a ∈ A (resp. b ∈ B ) the P -preimage of the G ′ -neighborhood P − (cid:2) G ′ ( a ) (cid:3) := P − (cid:2) { b ∈ B : ( a, b ) ∈ G ′ } (cid:3) . (resp. of G ′ ( b ) ) is ψ ( N, δ, K ) -separating.Furthermore, we will assume that the following technical conditions hold for φ ( N, δ, K ) , ψ ( N, δ, K ) :(A1) φ, ψ are non-decreasing in N (A2) φ is non-decreasing in δ , non-increasing in K and for each δ and K , we have φ ( N, δ, K ) ≤ N .(A3) ψ is non-decreasing in K (A4) If N ≥ M then φ ( N, δ, K ) N ≤ φ ( M, δ, K ) M Note that, by Claim 2.7, the pair ψ ( N, δ, K ) := N ; φ ( N, δ, K ) := δN is trivially admissible withmuch room to spare.The following lemma gives a Freiman-type pair of admissible functions which is better thantrivial in the regime K = o (log N ), and will be used later to bootstrap the argument. Lemma 6.2 (Freiman-type admissible functions) . There is an absolute constant C > such thatthe pair of functions(1) ψ ( N, δ, K ) := min (cid:26) (2 k )( Kδ ) C , N (cid:27) (2) φ ( N, δ, K ) := (cid:0) δK (cid:1) C N is admissible.Proof. This pair is easily seen to satisfy (A1) through (A4). Thus it remains to check (G) and (S).By the setup, we are given two sets A and B of sizes N A and N B respectively, and a graph G ofsize δN A N B such that | A + G B | ≤ K p N A N B (44)Assume without loss of generality that N A ≥ N B and take X = A ∪ B , which is of size ≈ N A .Since by (44) K δ N B ≥ N A we have | G | ≫ δ K | X | and | X + G X | ≪ K | C | . By a variant of the Balog-Szemer´edi-Gowers theorem (see e.g. [17], Exercise 6.4.10) there is X ′ ⊆ X such that | X ′ + X ′ | < K ′ | X ′ | and | G ∩ ( X ′ × X ′ ) | > δ ′ N A with δ ′ > (cid:18) δK (cid:19) C (45) K ′ < (cid:18) Kδ (cid:19) C . (46)By Theorem 2.9 any subset of X has rank at most K ′ and by Theorem 2.10, the P -preimage ofany subset of X ′ is at most (2 k ) K ′ C -separating for some C > 0. Thus, taking G ′ := G ∩ ( X ′ × X ′ )by (45) and (46) we verify that the pair (1), (2) is admissible. (cid:3) N ITERATED PRODUCT SETS WITH SHIFTS II 31 The goal is to find a better pair of admissible functions. The lemma below implements the‘induction on scales’ approach, which allows one to cook up a new pair φ ∗ ( N, · , · ) , ψ ∗ ( N, · , · ) from agiven pair of admissible functions, but taken at the smaller scale ≈ N / . Lemma 6.3. Let ψ and φ be an admissible pair of functions. Then for some absolute constant C > the pair of functions ψ ∗ ( N, δ, K ) := Ck max ψ ( N ′ , δ ′ , K ′ ) ψ ( N ′′ , δ ′′ , K ′′ ) (47) φ ∗ ( N, δ, K ) := min φ ( N ′ , δ ′ , K ′ ) φ ( N ′′ , δ ′′ , K ′′ ) (48) is admissible.Here min and max is taken over the data ( N ′ , δ ′ , K ′ ) , ( N ′′ , K ′′ , δ ′′ ) such that (cid:18) c δ log ( K/δ ) (cid:19) N ≤ N ′ N ′′ ≤ N (49) N ′ + N ′′ ≤ (cid:18) C K δ (cid:19) N / (50) K ′ K ′′ ≤ (cid:18) C log Kδ (cid:19) K (51) δ ′ δ ′′ ≥ (cid:18) c ( K/δ ) (cid:19) δ. (52) Proof. Let us first check that ( φ ∗ , ψ ∗ ) given by (47) and (48) indeed satisfy (A1) through (A4).Assume N < N and δ, K are fixed. Then ψ ∗ ( N , · , · ) < ψ ∗ ( N , · , · ) since for ψ ∗ ( N , · , · ) themaximum is taken over the larger range of parameters N ′ N ′′ ≤ N , N ′ + N ′′ ≤ Cδ − K N / . Similarly, φ ∗ ( N , · , · ) < φ ∗ ( N , · , · )since the minimum is now taken over the smaller set cδ log − ( K/δ ) N ≤ N ′ N ′′ . Note, that here we have used the fact that φ and ψ are both increasing. This proves (A1).In order to prove (A2) it suffices to note that when δ increases (resp. K decreases) the range ofparameters N ′ , N ′′ , δ ′ , δ ′′ , K ′ , K ′′ over which the minimum in φ ∗ is taken is getting more narrow.Similarly, when K increases the maximum in ψ ∗ is taken over a larger set which proves (A3).It remains to verify (A4). Let M, δ, K be fixed and M ′ , M ′′ , δ ′ , δ ′′ , K ′ , K ′′ be such that the min-imum for φ ∗ ( M, δ, K ) in (49) is achieved. Let c > cM ′ , cM ′′ , δ ′ , δ ′′ , K ′ , K ′′ are in the admissible range for φ ∗ ( c M, δ, K ) so φ ∗ ( c M, δ, K ) ≤ φ ( cM ′ , δ ′ , K ′ ) φ ( cM ′′ , δ ′′ , K ′′ ) ≤ c φ ( M ′ , δ ′ , K ′ ) φ ( M ′′ , δ ′′ , K ′′ )= c φ ∗ ( M, δ, K ) . Taking c such that c M = N we get (A4).Let A, B ⊂ Z n of sizes N A , N B respectively, G ⊆ A × B and suppose that the conditions (1)-(3)are satisfied with parameters ( N, δ, K ) where N = N A N B . Our ultimate goal is to find a subgraphof G of size at least φ ( N ′ , δ ′ , K ′ ) φ ( N ′′ , δ ′′ , K ′′ )such that the P -preimage of any its neighbourhoods is Ck ψ ( N ′ , δ ′ , K ′ ) ψ ( N ′′ , δ ′′ , K ′′ ) − separating , for some N ′ , N ′′ , K ′ , K ′′ , δ ′ , δ ′′ satisfying (49). Once this is done, the proof will be complete. Inorder to achieve this goal, we will apply Lemma 5.2 and then use the hypothesis that the pair ψ, φ is admissible for much smaller sets.Define a function f ( t ) for 0 ≤ t ≤ n as f ( t ) = max ( a ,b ) ∈ π [ t ] ( A ) × π [ t ] ( B ) {| A ( a ) | + | B ( b ) |} , where π [ t ] is the projection onto the first t coordinates, and A ( a ) and B ( b ) are the fibres above a and b respectively. Note that f is decreasing, f (0) = | A | + | B | ≥ N / , and f ( n ) = 0. Thusthere is t ′ such that f ( t ′ ) ≥ N / (53)but f ( t ′ + 1) < N / . (54)We use the t ′ defined above for the decomposition Z n = Z t ′ × Z n − t ′ and let π and π denotethe projection onto the first and second factor respectively. We now apply Lemma 5.2 and get sets A ′ ⊆ A and B ′ ⊆ B together with a graph G ′ ⊆ G ∩ ( A ′ × B ′ ) such that A ′ = [ a ∈ π ( A ′ ) a ⊕ A ′ ( a ) (55) B ′ = [ b ∈ π ( B ′ ) b ⊕ B ′ ( b ) (56)and the fibers A ′ ( a ) , B ′ ( b ) together with the fiber graphs G ′ ( a , b ) are uniform as defined in thestatement of Lemma 5.2. Note that it is possible that t ′ = 0, in which case the sets split triviallywith π ( A ′ ) = π ( B ′ ) = { } .Using the notation of Lemma 5.2 we have | π ( A ′ ) + G ′ π ( B ) | ≤ K ( M A M B ) / . (57)Since φ, ψ is an admissible pair, there is G ′′ ⊆ G ′ of size at least φ ( M M , δ , K ) such that all P -preimages of its vertex neighbourhoods are ψ ( M M , δ , K )-separating. Next, since G ′′ ⊆ G ′ ,for each edge ( a , b ) ∈ G ′′ , there is a graph G ′ ( a , b ) ⊆ A ′ ( a ) × B ′ ( b ) such that | G ′ ( a , b ) | ≥ δ m A m B and | A ′ ( a ) + G ′ ( a ,b ) B ′ ( b ) | ≤ K ( m A m B ) / . (58) N ITERATED PRODUCT SETS WITH SHIFTS II 33 Again, by admissibility of φ, ψ , there is G ′′ ( a , b ) ⊆ G ′ ( a , b ) of size at least φ ( m A m B , δ , K )such that all P -preimages of its vertex neighbourhoods are ψ ( m A m B , δ , K )-separating.Now define G ′′ ⊆ G ∩ ( A ′ × B ′ ) as G ′′ := { ( a ⊕ a , b ⊕ b ) : ( a , a ) ∈ G ′′ , ( a , b ) ∈ G ′′ ( a , b ) } . It is clear by construction that indeed all vertices of G ′′ belong to A ′ and B ′ respectively. Moreover,we have | G ′′ | ≥ φ ( M A M B , δ , K ) φ ( m A m B , δ , K ) . (59)Now let’s estimate the separating constant for the P -preimage of a neighbourhood P − [ G ′′ ( u )]of some u ∈ V ( G ′′ ). Without loss of generality assume that n ∈ B ′ and b = b ⊕ b . We can write G ′′ ( b ) = [ a ∈ G ′′ ( b ) [ a ∈ G ′′ ( a ,b ) { a ⊕ a } . (60)Thus, P − [ G ′′ ( b )] = [ a ∈ G ′′ ( b ) p a · [ a ∈ G ′′ ( a ,b ) p a . (61)Here we are using the notation q r = q r · · · q r l l for a vector q of primes and a vector r of integers,and p and p are respectively the first t primes from the map P and the remaining primes. Now,since G ′′ ( b ) and G ′′ ( a , b ) are orthogonal as linear sets we conclude that ( p a , p a ) = 1. Thus, byLemma 2.13 and the admissibility of φ, ψ applied to G ′′ and G ′′ ( a , b ) we conclude that P − [ G ′′ ( b )]is at most ψ ( M A M B , δ , K ) ψ ( m A m B , δ , K )-separating.We now record the bounds for the various parameters following from Lemma 5.2. We have δ δ ≥ (cid:18) c ( K/δ ) (cid:19) δ. (62) K K ≤ (cid:18) C log Kδ (cid:19) K (63) M A m A ≥ (cid:18) c δ log( K/δ ) (cid:19) N A (64) M B m B ≥ (cid:18) c δ log( K/δ ) (cid:19) N B (65) m A , m B ≥ (cid:18) c δ K (cid:19) N / (66)In particular, we have M A M B < N A N B m A m B < (cid:18) c K δ (cid:19) N / . (67)As a first attempt, we set N ′ = M A M B and N ′′ = m A m B , δ ′ = δ , K ′ = K , δ ′′ = δ and K ′′ = K .If N ′′ = m A m B is less than N / , one can verify that all of the above bounds comply with thestatement of this lemma, and we can stop. If N ′′ is too big, we will apply Lemma 5.2 again. To further reduce the size we apply Lemma 5.2 again for each pair of sets ( A ′ ( a ) , B ′ ( b )) suchthat ( a , b ) ∈ G ′ , stripping off only a single coordinate as explained below. Assume the base point( a , b ) is fixed henceforth.We split the coordinates { t ′ + 1 , . . . , n } as Z × Z n − t ′ − . We apply Lemma 5.2, this time with tothe pair of sets A ′ ( a ) and B ′ ( b ) and the graph G ′ ( a , b ). To ease notation, let us set U = A ′ ( a ), V = B ′ ( b ), and H = G ′ ( a , b ). Here, it is worth noting that U, V and H depend on the basepoint ( a , b ). This time, we have the estimates | U | ≈ m A , | V | ≈ m B and | U + H V | ≤ K ( | U || V | ) / where | H | ≥ δ | U || V | . We will again denote by π the projection onto the first coordinate, and by π the projection onto the remaining n − t ′ − U ′ ⊆ U, V ′ ⊆ V such that U ′ = [ u ∈ π ( U ′ ) u ⊕ U ′ ( u ) (68) V ′ = [ y ∈ π ( V ′ ) v ⊕ V ′ ( v ) (69)and the fibers U ′ ( u ) and V ′ ( v ) are of approximately the same size, say m U and m V respectively.We also write M U = | π ( U ) | and M V = | π ( V ) | . Note again that, for instance, the fiber U ′ ( u )may be trivial (i.e. { } ), which simply means that m U ≈ 1. By (28), (29) we have the estimates M U m U ≥ cδ (log( K /δ )) − | U | , M V m V ≥ cδ (log( K /δ )) − | V | Next, we have a graph H ′ ⊆ ( U ′ × V ′ ) ∩ H with uniform fibers as defined in Lemma 5.2. The graph H ′ splits into the base graph H ′ ⊂ π ( U ′ ) × π ( V ′ ) such that | π ( U ′ ) + H ′ π ( V ′ ) | ≤ K ( M U M V ) / , and fiber graphs H ′ ( u , v ) such that for ( u , v ) ∈ H ′ | U ′ ( u ) + H ′ ( u ,v ) V ′ ( v ) | ≤ K ( m U m V ) / , (70)with | U ′ ( u ) | ≈ m U (71) | V ′ ( v ) | ≈ m V (72) | H ′ ( u , v ) | ≥ δ m U m V . (73) N ITERATED PRODUCT SETS WITH SHIFTS II 35 The parameters m U , m V , δ , δ , K , K as well as the sizes of H ′ and H ′ ( u , v ) are controlledby Lemma 5.2. By the assumption that the original pair ( φ, ψ ) is admissible, for each such a graph H ′ ( u , v ) there is a subgraph H ′′ ( u , v ) ⊆ H ′ ( u , v ) with | H ′′ ( u , v ) | ≥ φ ( m U m V , δ , K ) (74)such that the P -preimage of each neighborhood of H ′′ ( u , v ) is ψ ( m U m V , δ , K )-separating. De-fine H ′′ ⊂ H ′ as H ′′ = { ( u ⊕ u , v ⊕ v ) : ( u , v ) ∈ H ′ , ( u , v ) ∈ H ′′ ( u , v ) } . (75)The size of H ′′ is at least | H ′ | φ ( m U m V , δ , K ). Next, the set of vertices of H ′ all lie in a one-dimensional affine subspace, so combining Corollary 2.8 and Lemma 2.13 one concludes that the P -preimage of each neighborhood of H ′′ is Ck ψ ( m U m V , δ , K )-separating with some absoluteconstant C > 0. Putting together all of the details, we conclude that, for G ′ ( a , b ) ⊂ A ′ ( a ) × B ′ ( b ), there is a subgraph H ′′ ⊆ G ′ ( a , b ) of size at least φ a ,b := | H ′ | φ ( m U m V , δ , K ) (76)such that the P -preimage of each neighbourhood in H ′′ is ψ a ,b -separating, where ψ a ,b := Ck ψ ( m U m V , δ , K ) . Since the the graph H ′′ depends on the pair ( a , b ), we now rename this graph H ′′ a ,b .In turn, substituting ψ a ,b and φ a ,b into the argument leading to (59) and Lemma 2.13, weconstruct a graph G ′′′ := { ( a ⊕ a , b ⊕ b ) : ( a , b ) ∈ G ′ , ( a , b ) ∈ H ′′ a ,b } . The graph G ′′′ has size at least φ ( M A M B , δ , K ) · min ( a ,b ) ∈ G ′ φ a ,b , (77)and the separating factors are at most ψ ( M A M B , δ , K ) · max ( a ,b ) ∈ G ′ ψ a ,b , (78)With G ′′′ we have now found a large subgraph with good separating factors. In the remainingcalculations, we show that the existence of this G ′′′ is good enough to imply the theorem. Essentiallyit remains to check that the quantities (77) and (78) can indeed be bounded respectively by (48) and(47). Note that the quantities (77) and (78) do depend on the structure of A and B . We are going toshow, however, that they are uniformly bounded by (48) and (47) which are functions of ( N, δ, K )only. We remark here that we will make use of the following fact: if | X + G Y | ≤ K ( | X || Y | ) / forsome G ⊂ X × Y of size at least δ | X || Y | , then K/δ ≥ a , b ) ∈ G ′ we have by (32) δ ≥ δ δ > c log − ( K /δ ) δ . (79) By (35) and (32) K δ ≤ K K δ δ < CK log( K ) log ( K/δ ) δ (80)and so log( K /δ ) < C log( K/δ ) . (81)Consequently, δ δ , (81) > c log − ( K/δ ) δ δ > c log − ( K/δ ) δ. (82)Next, by (35) K ≤ K K δ δ ≤ CK log ( K ) δ − (83)and by (32) δ > c log − ( K/δ ) δ (84) K < Cδ − K log K. (85)Therefore log ( K ) δ − ≤ C log ( K/δ ) δ − (86)= C ( δ log ( K/δ )) δ − < C (log K ) δ − and K K ≤ CK K log ( K ) δ − 42 ( ) , ( ) ≤ C K log Kδ . (87)Finally, we have by (28), (32), (33) and (34) that | H ′ | m U m V ≥ c log − ( K /δ ) δ ( δ log − ( K /δ )) | A ( a ) || B ( b ) |≥ c log − ( K/δ ) δ m A m B ( ) ≥ c log − ( K/δ ) δ m A m B . (88)Define N ′′ := min { N / , max { m U m V , c log − ( K/δ ) δ m A m B }} . (89)By our choice of t ′ it follows that m U m V ≤ N ′′ . By (A4) we have m U m V N ′′ φ ( N ′′ , δ , K ) ≤ φ ( m U m V , δ , K ) . (90)Defining N ′ := M A M B m U m V N ′′ | H ′ | , (91)we have by (88) and (89) that M A M B ≤ N ′ , so by (A4) again M A M B N ′ φ ( N ′ , δ , K ) ≤ φ ( M A M B , δ , K ) , (92) N ITERATED PRODUCT SETS WITH SHIFTS II 37 so φ ( N ′ , δ , K ) φ ( N ′′ , δ , K ) ≤ N ′ M A M B φ ( M A M B , δ , K ) N ′′ m U m V φ ( m U m V , δ , K ) ( ) = φ ( M A M B , δ , K ) φ a ,b . (93)On the other hand, N ′ N ′′ = M A M B m U m V | H ′ | (94) ( ) ≥ c log − ( K/δ ) δ M A M B m A m B (95) ( ) , ( ) ≥ cδ log − ( K/δ ) N. (96)Also, since m A m B (66) > cδ K − N / , it follows from the definition of N ′′ in (89) that cδ K − N / ≤ N ′′ ≤ N / . Then, since N ′′ N ′ ≤ N , N ′ ≤ Cδ − K N / , and so N ′ + N ′′ ≤ Cδ − K N / . (97)We now have all the estimates to finish the proof. The bounds (82), (87), (94), (97) verify thatthe parameters δ ′ := δ , δ ′′ := δ K ′ := K , K ′′ := K (98)and N ′ , N ′′ indeed satisfy the constraints (49). Recall that by (A1) ψ ( · , δ, K ) is increasing in thefirst argument, so by (67) and (54) ψ ∗ ( N, δ, K ) ≥ Ck ψ (cid:18) max (cid:26) N / , Nm A m B (cid:27) , δ , K (cid:19) ψ (cid:16) min { N / , m A m B } , δ , K (cid:17) ≥ ψ ( M A M B , δ , K ) ψ x,y . (99)In the previous inequality, we have used monotonicity (A1) and the information that Nm A m B ≥ M A M B , N / ≥ m U m V , m A ≥ m U and m B ≥ m V .Also, (93) and (77) verify that φ ∗ ( N, δ, K ) ≤ φ ( N ′ , δ , K ) φ ( N ′′ , δ , K ) ≤ φ ( M A M B , δ , K ) φ a ,b . (100) It follows that the pair ( ψ ∗ , φ ∗ ) is indeed admissible since (99) and (100) hold for all base points( a , b ) ∈ G ′ and thus uniformly bound (78) and (77) respectively. (cid:3) A better admissible pair With Lemma 6.3 at our disposal we can start with the data ( N, δ, K ) and reduce the problem tothe case of smaller and smaller N and K with reasonable losses in δ . The process can be describedby a binary a tree where each node with the data ( N, δ, K ) splits into two children with the attacheddata being approximately equal to ( N / , δ ′ , K ′ ) and ( N / , δ ′ , K ′′ ), with K ′ K ′′ roughly equal to K and δ ′ δ ′′ roughly equal to δ . Thus, when the height of the tree is about log log K , the K ’s in themost of the nodes should be small enough so that Lemma 6.2 becomes non-trivial. Going from thebottom to the top we then recover an improved admissible pair of functions at the root node. Lemma 7.1. For any γ > there exists C ( γ ) > such that the pair φ ( N, δ, K ) := (cid:18) δK (cid:19) C log log( K/δ ) N (101) ψ ( N, δ, K ) := k log( K/δ ) C/γ N γ (102) is admissible.Proof. Let N, δ, K be fixed. Take an integer t = 2 l to be specified later ( l is going to be the heightof the tree and t the total number of nodes).Let ( φ , ψ ) be the Freiman-type admissible pair given by Lemma 6.2. We apply recursivelyLemma 6.3 and obtain admissible pairs for i = 1 , . . . , l as follows ψ i := max Ck ψ i − ( N ′ , δ ′ , K ′ ) ψ i − ( N ′′ , δ ′′ , K ′′ ) (103) φ i := min φ i − ( N ′ , δ ′ , K ′ ) φ i − ( N ′′ , δ ′′ , K ′′ ) , (104)(with the max and min taken over the set of parameters constrained by (49)). Thus, at the rootnode we have the admissible pair ψ := ψ l − , φ := φ l − given by ψ ( N, δ, K ) := ( Ck ) l Y ν ∈{ , } l ψ ( N ′ ν , δ ′ ν , K ′ ν ) (105) φ ( N, δ, K ) := Y ν ∈{ , } l φ ( N ν , δ ν , K ν ) (106)for some data ( N ν , δ ν , K ν ) and (possibly different) ( N ′ ν , δ ′ ν , K ′ ν ) at the leaf nodes of the tree whichattain the respective maxima and minima. For intermediate tree nodes ν , denoting by { ν, } and N ITERATED PRODUCT SETS WITH SHIFTS II 39 { ν, } the left and right child of ν respectively, one has c δ ν log − ( K ν /δ ν ) N ν ≤ N ν, N ν, ≤ N ν (107) N ν, + N ν, ≤ C δ − ν K ν N / ν (108) K ν, K ν, ≤ C log K ν δ ν K ν (109) δ ν, δ ν, ≥ c log − ( K ν /δ ν ) δ ν , (110)and similarly for ( N ′ ν , δ ′ ν , K ′ ν ). The absolute constants c and C are exactly those given in thestatement of Lemma 6.3 as c and C respectively. They have been relabelled here in an attempt todistinguish them.In what follows we assume that N is large enough so that log K ν > C and log( δ − ν ) > c − andthe constants C, c can be swallowed by an extra power of log( K/δ ).We have log K ν, δ ν, + log K ν, δ ν, < 20 log K ν δ ν so for an arbitrary 1 < l ′ ≤ l max ν ∈{ , } l ′ log K ν δ ν ≤ X ν ∈{ , } l ′ log K ν δ ν < l ′ log Kδ . (111)Next, it follows from (110) and (111) that Y ν ∈{ , } l ′ δ ν = Y ν ∈{ , } l ′− δ ν, δ ν, ≥ Y ν ∈{ , } l ′− c (cid:18) log K ν δ ν (cid:19) − δ ν > Y ν ∈{ , } l ′− c (cid:18) l ′ log Kδ (cid:19) − Y ν ∈{ , } l ′− δ ν = (cid:18) c (cid:19) − l ′ · l ′ (cid:18) log Kδ (cid:19) − · l ′ Y ν ∈{ , } l ′− δ ν . (112)Applying (112) iteratively then yields Y ν ∈{ , } l ′ δ ν > (cid:18) c (cid:19) − l ′ · l ′ (cid:18) log Kδ (cid:19) − · l ′ δ. (113)Using similar arguments, we obtain the following bounds: Y ν ∈{ , } l ′ K ν < (cid:18) C c (cid:19) · l ′ l ′ (cid:18) log Kδ (cid:19) · l ′ δ − l ′ K (114)and Y ν ∈{ , } l ′ N ν > (cid:18) c (cid:19) − · l ′ l ′ (cid:18) log Kδ (cid:19) − · l ′ δ l ′ N. (115)For more details on how these bounds are obtained, see [4, p. 492]. Substituting (113), (114), (115) into (106) and Lemma 6.2 (2) we get φ ( N, δ, K ) = Y ν ∈{ , } l φ ( N ν , δ ν , K ν ) = Y ν ∈{ , } l (cid:18) δ ν K ν (cid:19) C N ν ≥ e − C ′ l l (cid:18) log Kδ (cid:19) − C ′ l δ lC ′ K − C ′ N, for some suitable C ′ > 0. Taking l := log log( K/δ )we obtain φ ( N, δ, K ) ≥ (cid:18) δK (cid:19) C log log( K/δ ) N for some suitable C > ψ . For the sake of notation we use again ( N ν , δ ν , K ν ) instead of ( N ′ ν , δ ′ ν , K ′ ν ).The bounds above, however, still hold.By (105) and Lemma 6.2 ψ ( N, δ, K ) = ( Ck ) l Y ν ∈{ , } l min (cid:26) (2 k )( Kνδν ) C , N ν (cid:27) . (116)In order to bound the quantity of the right hand side effectively, we will need a suitable uniformbound for individual N ν , which we deduce below.It follows from (110) that δ ν, , δ ν, ≥ c (cid:18) log K ν δ ν (cid:19) − δ ν . (117)Applying this bound as well as (111), it follows that for any 1 ≤ l ′ ≤ l and ν ∈ { , } l ′ , δ ν = δ ν ′ , · ≥ c (cid:18) log K ν ′ δ ν ′ (cid:19) − δ ν ′ ≥ (20 C ) − l ′ (cid:18) log Kδ (cid:19) − δ ν ′ . (118)Iteratively applying (118) yields δ ν ≥ (20 C ) − l ′ (cid:18) log Kδ (cid:19) − l ′ δ. (119)Similarly, since K ν ≥ δ ν , it follows from (109) and (117) that K ν, ≤ C K ν log K ν δ ν δ ν, ≤ C ′ (cid:16) log K ν δ ν (cid:17) δ ν K ν . The same argument implies that K ν, ≤ C ′ K ν (cid:16) log Kνδν (cid:17) δ ν . Therefore, by applying (111) and (119),it follows that for any ν ∈ { , } l ′ , K ν = K ν ′ , ∗ ≤ C ′ K ν ′ (cid:16) log K ν ′ δ ν ′ (cid:17) δ ν ′ ≤ (20 C ) l ′ (cid:0) log Kδ (cid:1) l ′ δ K ν ′ . (120) Strictly speaking we should ensure that l is an integer by taking l := ⌊ log log( K/δ ) ⌋ . In order to simplifycalculations and avoid adding further multiplicative constants, we assume that l as defined here is already an integer. N ITERATED PRODUCT SETS WITH SHIFTS II 41 Iterating (120) yields K ν ≤ (20 C ) l ′ (cid:0) log Kδ (cid:1) l ′ δ l ′ K. (121)To bound N ν , first note that (108), (119) and (121) together imply that for any ν ′ ∈ { , } l ′ , N ν, + N ν, ≤ C δ − ν K ν N / ν ≤ (20 C ) l ′ (cid:0) log Kδ (cid:1) l ′ K δ l ′ N / ν . Applying this bound iteratively yields (with some rather crude estimates) N ν ≤ (20 C ) l ′ (cid:0) log Kδ (cid:1) l ′ K δ l ′ N l ′ . (122)Before inserting (122) into (116), we split the data ( N ν , δ ν , K ν ) into two parts, I ∪ J = { , } l ,such that I = (cid:26) ν : K ν δ ν < T (cid:27) and J = (cid:26) ν : K ν δ ν ≥ T (cid:27) , with the threshold T specified later.By (113) and (114) we see that | J | is rather small: T | J | ≤ Y ν ∈{ , } l K ν δ ν < (cid:18) C c (cid:19) · l l log( Kδ ) · l δ − l K. (123)Set t := 2 l , so it follows from (123) that for an appropriate constant C , | J | log T ≤ C lt. Choose log T := C γ − l = C log log( K/δ ) γ . (124)Thus | J | t ≤ C l log T = γ. (125)We are finally ready to put everything together: ψ ( N, δ, K ) (116) = ( Ck ) l Y ν ∈{ , } l min (cid:26) (2 k )( Kνδν ) C , N ν (cid:27) . ≤ ( Ck ) l Y ν ∈ I (2 k ) T C Y ν ∈ J N ν (122) ≤ ( C ′ k ) tT C (20 C ) l (cid:0) log Kδ (cid:1) l K δ l N t | J | (125) ≤ k ( log Kδ ) C ′′ γ N γ . (cid:3) A strong admissible pair Finally, in this section we will use Lemma 7.1 to get an even better pair of admissible functions. Lemma 8.1. Given < τ, γ < / there exist positive constants α i ( τ, γ, k ) , β i ( τ, γ, k ) , i = 1 , , such that for all sufficiently large N , the pair φ ( N, δ, K ) := K − α δ α log log N e α (log log N ) N − τ (126) ψ ( N, δ, K ) := K β δ − β log log N e − β (log log N ) N γ (127) is admissible.Proof. The strategy of the proof is as follows. We start with the already not-so-bad admissible pairgiven by Lemma 7.1 and improve it by repeated application of Lemma 6.3.Let P N [ φ, ψ ] be the predicate that the pair ( φ, ψ ) given by (126) and (127) is admissible in thesense of Definition 6.1 for all graphs of size at most N and at least N / .We are going to prove that(1) The base case: P N [ φ, ψ ] is true for some N ( τ, γ ).(2) The inductive step: P N [ φ, ψ ] ⇒ P N / [ φ, ψ ].The exponent 3 / N ≥ N .In order to prove (1) it suffices to find a fixed threshold N ( τ, γ ) such that the pair (126), (127) iseither trivial or worse than that given by Lemma 7.1 if N ≤ N . One can achieve this by fine-tuningthe constants α , β , which we now explain.Apply Lemma 7.1 with γ = γ/ α , β and N ( δ, γ ) such that for each N in the range N / ≤ N ≤ N (cid:18) δK (cid:19) C ( γ ) log log( K/δ ) N ≥ K − α δ α log log N e α (log log N ) N − τ (128)min { N, exp (cid:16) log k · log( K/δ ) C ( γ ) /γ (cid:17) N γ/ } ≤ K β δ − β log log N e − β (log log N ) N γ . (129)To ensure (128) holds it is sufficient to take α = C ( γ )2 log log N with C ( γ ) > α = C α and α = C α for some absolute constants C , C ≥ 1. Indeed, K − α δ α log log N e α (log log N ) N − τ ≤ (cid:18) δK (cid:19) α e α (log log N ) N − τ ≤ (cid:18) δK (cid:19) C ( γ ) log log N e α (log log N ) N − τ ≤ (cid:18) δK (cid:19) C ( γ ) log log N N, where the last inequality holds as long as we take N sufficiently large (and thus also N is sufficientlylarge). Inequality (128) then follows since the inequality N ≥ Kδ holds by definition of N, δ and K . N ITERATED PRODUCT SETS WITH SHIFTS II 43 Ensuring (129) is more involved, as later on want to impose the further constraint β > β > β .For now, it suffices to guarantee thatlog k · log (cid:18) Kδ (cid:19) Cγ < γ N (130)and e β (log log N ) < N γ . (131)However, the bound (130) fails only if K/δ is rather large, namely Kδ > e log cγ N for some c ( C, γ, k ) > 0. In this case it suffices to take β so large that K β δ − β log log N e − β (log log N ) N γ > N and thus (129) holds. To this end, we set β := (log N ) − cγ and make the constraint that, say, β , β < β log log N . Moreover, this constraint on β also ensures that (131) holds for N sufficiently large.Summing up, we have found some fixed threshold N ( τ, γ ) at which (126), (127) become admis-sible with fixed α , β and still some freedom to define the constants α , β , α , and β .We now turn to part (2) of the induction scheme, the inductive step. Assuming that N ′ , N ′′ areat the scale so that (126), (127) are admissible with the data ( N ′ , δ ′ , K ′ ); ( N ′′ , δ ′′ , K ′′ ) we will showthat (126), (127) are also admissible for the data ( N, δ, K ) with N ≈ N ′ N ′′ .Assuming β (or N ) is large enough we may assume that Kδ < N − , (132)as otherwise (127) > N which is trivially admissible.We need to estimate ψ ( N ′ , δ ′ , K ′ ) ψ ( N ′′ , δ ′′ , K ′′ )from above and φ ( N ′ , δ ′ , K ′ ) φ ( N ′′ , δ ′′ , K ′′ ) , from below in order to verify that (126), (127) are admissible for ( N, δ, K ). By (132), the constraints(49) can be relaxed to N ≥ N ′ N ′′ > N (cid:18) δ log N (cid:19) > N / (133) N ′ + N ′′ < N / (cid:18) Kδ (cid:19) < N / / (134) δ ′ δ ′′ > δ log N (135) K ′ K ′′ < δ − (log N ) K. (136)From (133) and (134) we have (with room to spare) N / − / < N ′ , N ′′ < N / / (137)and so assuming N is large enough99100 log log N < log log N ′ , log log N ′′ < log log N − log 2011 . (138)With the constraints above, it suffices to verify (writing ll for log log as in [4]) that( K ′ K ′′ ) − α ( δ ′ ) α llN ′ ( δ ′′ ) α llN ′′ e α [( llN ′ ) +( llN ′′ ) ] ( N ′ N ′′ ) − τ (139)is indeed always bounded below by (126). We can bound (139) by K − α δ α llN e α ( llN ) N − τ u · v (140)where u = (log N ) − α − α llN − e α ( llN ) (141) v = δ α − log α +40 . (142)For suitable choices of α , α > α both u, v > K ′ K ′′ ) β ( δ ′ ) − β llN ′ ( δ ′′ ) − β llN ′′ e − β [( llN ′ ) +( llN ′′ ) ] ( N ′ N ′′ ) γ (143) < K β δ − β llN e − β ( llN ) N γ u · v (144)with u = (log N ) β +6 β llN e − β ( llN ) (145) v = δ − β +log β . (146)Again, by taking suitable β > β > β we make u, v < (cid:3) N ITERATED PRODUCT SETS WITH SHIFTS II 45 Concluding the proof of Theorem 3.1 We are finally ready to finish the proof of Theorem 3.1. Recall that the aim is to show that,given 0 < τ, γ < / 2, there are positive constants C = C ( τ, γ, k ) and C = C ( τ, γ, k ), such thatfor any A ⊂ Q with | AA | ≤ K | A | , there exists A ′ ⊂ A with | A ′ | ≥ K − C | A | − τ , such that A ′ is K C | A | γ -separating.Since | AA | ≤ K | A | , after applying the prime evaluation map, we have |P ( A )+ P ( A ) | ≤ K |P ( A ) | .Fix γ ′ = γ/ τ ′ = τ / 2, and apply Lemma 8.1 for this choice of γ ′ , τ ′ , with the full graph G = P ( A ) × P ( A ). It follows that there is a subgraph G ′ ⊂ G such | G ′ | ≥ K − α e α (log log | A | ) | A | − τ ′ ≥ K − α | A | − τ ′ and such that for each v ∈ V ( G ) the P -preimages of N G ′ ( v ) is K β e − β (log log | A | ) | A | γ ′ ≤ K β | A | γ ′ separating. Then, by the pigeonhole principle, there is a vertex v ∈ V ( G ) such that | N G ′ ( v ) | ≥ | A | − τ ′ .Write A ′ = P − ( N G ′ ( v )) for the preimage of the neighbourhood of v . Then this is a subset of A with the required properties. 10. Further Applications Proof of Theorem 1.8. Recall that Theorem 1.8 is the following statement. For all γ ≥ C = C ( γ ) such that for any finite A ⊂ Q with | AA | ≤ K | A | and any finite set L of lines in the plane, I ( P, L ) ≤ | P | + | A | γ K C | L | , where P = A × A .First of all, observe that horizontal and vertical lines contribute a total of at most 2 | P | . Thisis because each point p ∈ P can belong to at most one horizontal and one vertical line. Similarly,lines through the origin contribute at most | P | + | L | incidences, since each point aside from theorigin belongs to at most one such line, and the origin itself may contribute | L | incidences.It remains to bound incidences with lines of the form y = mx + c , with m, c = 0. Let l m,c denotethe line with equation y = mx + c . Note that, if m / ∈ Q then l m,c contains at most one point from P . Indeed, suppose l m,c contains two distinct points ( x, y ) and ( x ′ , y ′ ) from P . In particular, since A ⊂ Q , x, y, x ′ , y ′ ∈ Q . Then l m,c has direction m = y − y ′ x − x ′ . Therefore, lines l m,c with irrationalslope m contribute at most | L | incidences.Next, suppose that m ∈ Q and c / ∈ Q . Then l m,c does not contain any points from P , since if itdid then we would have a solution to y = mx + c , but the left hand side is rational and the righthand side is irrational.It remains to consider the case when m, c ∈ Q ∗ . An application of Theorem 1.7 implies that | l m,c ∩ P | ≤ K C | A | γ . Therefore, these lines contribute a total of at most | L | K C | A | γ incidences.Adding together the contributions from these different types of lines completes the proof. (cid:3) Note here that we have discarded the extra information coming from the terms of the form e ± C (log log | A | ) . Proof of Theorem 1.9. Recall that Theorem 1.9 states that, for any γ > C ( γ ) suchthat for an arbitrary A ⊂ Q with | AA | = K | A | and B, B ′ ⊂ Q , S := (cid:12)(cid:12) { ( b, b ′ ) ∈ B × B ′ : b + b ′ ∈ A } (cid:12)(cid:12) ≤ | A | γ K C min {| B | / | B ′ | + | B | , | B ′ | / | B | + | B ′ |} . We will prove that S ≤ | A | γ K C ( | B ′ | / | B | + | B ′ | ) . (147)Since the roles of B and B ′ are interchangeable, (147) also implies that S ≤ | A | γ K C ( | B | / | B ′ | + | B | ), and thus completes the proof.Let γ > C ( γ ), given by Theorem 1.7, be fixed. Without loss of generality assume that S ≥ | B ′ | as otherwise the claimed bound is trivial.For each b ∈ B define S b := { b ′ ∈ B ′ : b + b ′ ∈ A } , and similarly for b ′ ∈ B ′ T b ′ := { b ∈ B : b ′ + b ∈ A } . It follows from Theorem 1.7 that for b , b ∈ B with b = b | S b ∩ S b | ≤ | A | γ K C since each x ∈ S b ∩ S b gives a solution ( a, a ′ ) := ( b + x, b + x ) to a − a ′ = b − b with a, a ′ ∈ A .On the other hand, by double-counting and the Cauchy-Schwarz inequality, X b ∈ B | S b | + X b ,b ∈ B : b = b | S b ∩ S b | = X b ′ ∈ B ′ | T b ′ | ≥ | B ′ | − ( X b ′ ∈ B ′ | T b ′ | ) = | B ′ | − S . Therefore, X b ,b ∈ B : b = b | S b ∩ S b | ≥ | B ′ | − S − X b ∈ B | S b | = | B ′ | − S − S ≥ | B ′ | − S by our assumption.The left-hand side is at most | B | | A | γ K C , and so S ≤ (2 | A | γ K C ) / | C | / | B ′ | , which completes the proof. (cid:3) Proof of Theorem 1.10. Recall that Theorem 1.10 states that for all b there exists k such that forall A, B ⊂ Q with | B | ≥ | ( A + B ) k | ≥ | A | b .Since | B | ≥ 2, there exist two distinct elements b , b ∈ B . Apply Theorem 1.1 to conclude thatfor all b there exists k = k ( b ) with | ( A + B ) k | ≥ max {| ( A + b ) k | , | (( A + b ) + ( b − b )) k |} ≥ | A | b . (cid:3) N ITERATED PRODUCT SETS WITH SHIFTS II 47 Acknowledgements Oliver Roche-Newton was partially supported by the Austrian Science Fund FWF Project P30405-N32. Dmitrii Zhelezov was supported by the Knut and Alice Wallenberg Foundation Programfor Mathematics 2017.We thank Brendan Murphy, Imre Ruzsa and Endre Szemer´edi for helpful conversations. References [1] F. Amoroso and E. Viada, ‘Small points on subvarieties of a torus’, Duke Math. J. Electron. J. Combin. (2017), no. 3, Paper 3.14, 17 pp.[3] F. Beukers and H. P. Schlickewei, ‘The equation x + y = 1 in finitely generated groups’, Acta Arith. J. Amer. Math. Soc. ,no. 2, (2004), 473-497.[5] J. Bourgain and M.-C. Chang, ‘Sum-product theorems in algebraic number fields’ J. Anal. Math. (2009),253-277.[6] M.-C. 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