Variations on the sum-product problem II
aa r X i v : . [ m a t h . C O ] A p r VARIATIONS ON THE SUM-PRODUCT PROBLEM II
BRENDAN MURPHY, OLIVER ROCHE-NEWTON AND ILYA D. SHKREDOV
Abstract.
This paper is a sequel to a paper entitled
Variations on the sum-product problem by the same authors [7]. In this sequel, we quantitatively improve several of the main resultsof [7], as well as generalising a method from [7] to give a near-optimal bound for a newexpander.The main new results are the following bounds, which hold for any finite set A ⊂ R : ∃ a ∈ A such that | A ( A + a ) | & | A | + , | A ( A − A ) | & | A | + , | A ( A + A ) | & | A | + , |{ ( a + a + a + a ) + log a : a i ∈ A }| ≫ | A | log | A | . Introduction
Throughout this paper, the standard notation ≪ , ≫ is applied to positive quantities inthe usual way. Saying X ≫ Y means that X ≥ cY , for some absolute constant c >
0. Thenotation X ≈ Y denotes that X ≫ Y and X ≪ Y occur simultaneously. All logarithms inthe paper are base 2. We use the symbols . , & to suppress both constant and logarithmicfactors. To be precise, we write X & Y if there is some absolute constant c > X ≫ Y / (log X ) c .This paper considers several variations on the sum-product problem, all of which followa common theme. The story of the sum-product problem begins with the Erd˝os-Szemer´ediconjecture, which states that for any finite A ⊂ Z and for all ǫ > {| A + A | , | AA |} ≥ c ǫ | A | − ǫ , where A + A := { a + b : a, b ∈ A } is the sum set and AA := { ab : a, b ∈ A } is the product set . Although the conjecture was originally stated for subsets of the integers,it is also widely believed to be true over the reals.Modern literature on the sum-product problem often focuses on the problem of provingthat certain sets defined by a combination of different arithmetic operations on elements of a set A are always significantly larger than | A | . Growth results of this type are often referredto as expanders .In [7], the authors considered several expander problems. The aim of this sequel is toimprove the main results from [7].One result that was established in [7] is that, for any A ⊂ R ,(1.1) | A ( A + A ) | & | A | + . This result gave a small quantitative improvement on the inequality | A ( A + A ) | ≫ | A | / ,which follows from a simple application of the Szemer´edi-Trotter Theorem (see [10, Lemma3.2] for a formal proof).The exponent 3 / / | AA + A | ≫ | A | / , and no improvement is known.The main new theorem that we prove in this paper is another example of a result thatbreaks the 3 / Theorem 1.
Let A ⊂ R be finite. Then, there exists a ∈ A such that | A ( A + a ) | & | A | + . This improves on [7, Theorem 2.9], in which it was established that there is some a ∈ A such that | A ( A + a ) | ≫ | A | / . Sketch of the proof of Theorem 1.
There are two main new lemmas which go intothe proof of Theorem 1. The first of these is a lemma which states that there exists a ∈ A such that | A ( A + a ) | ≥ | A | + c , where c > of A is essentially as large as possible. This lemma is proved using the Szemer´edi-Trotter Theorem. In proving such a result, we improve qualitatively and quantitatively onsome of the main lemmas from [7]. See the forthcoming Lemma 9.The second new lemma (see the forthcoming Lemma 11 for a more precise statement)proves that if the product set of A is very small then the bound | A ( A + α ) | ≫ | A | + c holdsfor any non-zero α ∈ R . The proof of this is a little more involved, using techniques fromadditive combinatorics, and is closely related to the work of the third author in [14]. A non-trivial result bounding the additive energy for sets with small product set, also due to the The multiplicative energy of A is the number of solutions to the equation a a = a a , ( a , a , a , a ) ∈ A. See section 2 for more on the multiplicative energy and other types of energy.
ARIATIONS ON THE SUM-PRODUCT PROBLEM II 3 third author in [13], is used as a black box in the proof of this lemma. See the forthcomingTheorem 13 for the statement.The Balog-Szemer´edi-Gowers Theorem tells us that if a set A has large multiplicativeenergy, then there is a large subset A ′ ⊂ A such that | A ′ A ′ | is small. Therefore, one canthen use the Balog-Szemer´edi-Gowers Theorem and combine these two lemmas together toconclude the proof. However, we instead use a technical argument, building on the work ofKonyagin and Shkredov [3], which allows us to avoid an application of the Balog-Szemer´edi-Gowers Theorem and thus to improve the aforementioned constant c .1.2. Further new results.
The proof of inequality (1.1) in [7] followed a similar structureto the above sketch, and the Balog-Szemer´edi-Gowers Theorem was used to conclude theargument. Once again, we are able to make this argument more efficient by using toolsfrom [3] to avoid using the Balog-Szemer´edi-Gowers Theorem, resulting in the following tworesults.
Theorem 2.
Let A ⊂ R be finite. Then, | A ( A + A ) | & | A | + . Theorem 3.
Let A ⊂ R be finite. Then, | A ( A − A ) | & | A | + . Theorem 2 gives an improvement on (1.1), while Theorem 3 gives an improvement onequation (49) in [7].Another of the main results in [7] was the bound(1.2) | A ( A + A + A + A ) | ≫ | A | log | A | . Note that (1.2) is optimal, up to finding the correct power of the logarithmic factor, as canbe seen by taking A = { , , . . . , N } . In the last of our new theorems, we follow a similarargument to prove the following, admittedly curious, expander bound. Theorem 4.
Let A ⊂ R + be finite. Then, |{ ( a + a + a + a ) + log a : a i ∈ A }| ≫ | A | log | A | . Note that the simple example whereby A = { , , . . . , N } illustrates that Theorem 4 isalso optimal up to the logarithmic factor.1.3. The structure of the rest of this paper.
The rest of the paper will be structuredas follows. Section 2 will be used to introduce some notation and preliminary results thatwill be used throughout the paper. As mentioned above, there are two main new lemmasin this paper. Section 3 is devoted to proving the first of these, and section 4 the second.Section 5 is used to conclude the proof of Theorem 1. In section 6 the proofs of Theorem 2and 3 are concluded. Section 7 is devoted to the proof of Theorem 4.
B. MURPHY, O. ROCHE-NEWTON AND I. SHKREDOV
A note on an earlier preprint [9] . This paper supercedes the preprint [9] by thesecond author. 2.
Notation and preliminary results
Given finite sets
A, B ⊂ R , the additive energy of A and B is the number of solutions tothe equation a − b = a − b The additive energy is denoted E + ( A, B ). Let r A − B ( x ) := |{ ( a, b ) ∈ A × B : a − b = x }| . Note that r A − B ( x ) = | A ∩ ( B + x ) | . The notation of the representation function r will be usedwith flexibility throughout this paper, with the information about the kind of representationsit counts being contained in a subscript. For example, r ( A − A ) +( A − A ) ( x ) = |{ ( a , a , a , a ) ∈ A : ( a − a ) + ( a − a ) = x }| . Note that E + ( A, B ) = X x ∈ A − B r A − B ( x ) . The shorthand E + ( A ) = E + ( A, A ) is used.The notion of energy can be extended to an arbitrary power k . We define E + k ( A ) by theformula E + k ( A ) = X x ∈ A − A r kA − A ( x ) . Similarly, the multiplicative energy of A and B , denoted E × ( A, B ), is the number of solutionsto the equation a b = a b such that a , a ∈ A and b , b ∈ B . For x = 0, let A x denote the set A x = A ∩ x − A andnote that r A/A ( x ) = | A x | .A simple but important feature of energies is that the Cauchy-Schwarz inequality can beused to convert an upper bound for energy into a lower bound for the cardinality of a set.In particular, it follows from the Cauchy-Schwarz inequality that(2.1) E × ( A, B ) ≥ | A | | B | | AB | . The notions of additive and multiplicative energy have been central in the literature onsum-product estimates. For example, the key ingredient in the beautiful work of Solymosi[15], which until recently held the record for the best known sum-product estimate, is thefollowing bound:
Theorem 5.
For any finite A ⊂ R , E × ( A ) ≪ | A + A | log | A | . ARIATIONS ON THE SUM-PRODUCT PROBLEM II 5
A major tool that is used in this paper several times, both explicitly and implicitly, is theSzemer´edi-Trotter Theorem. In particular, we will need the following result, which followsfrom a simple application of the Szemer´edi-Trotter Theorem. See, for example, Corollary8.8 in [16].
Lemma 6.
Let A ⊂ R be a finite set. Then there are O ( | A | log | A | ) collinear triples in A × A . In a recent paper of Konyagin and Shkredov [3], a new characteristic for a finite set ofreal numbers A was considered. Define d ∗ ( A ) by the formula d ∗ ( A ) = min t> min ∅6 = Q,R ⊂ R \{ } | Q | | R | | A | t , where the second minimum is taken over all Q and R such that max {| Q | , | R |} ≥ | A | andsuch that for every a ∈ A , the bound | Q ∩ aR − | ≥ t holds. Konyagin and Shkredov provedthe following lemma: Lemma 7 (Lemma 13, [3]) . For any
A, B ⊂ R and any τ ≥ |{ x : r A − B ( x ) ≥ τ }| ≪ | A || B | τ d ∗ ( A ) . To put this in the language introduced in [12] and also used in [3], this lemma states thatevery set A is a Szemer´edi-Trotter set with parameter O ( d ∗ ( A )). The main theoretical toolin the proof of Lemma 7 is the Szemer´edi-Trotter Theorem. Lemma 7 generalises an earlierresult in which the bound(2.2) |{ x : r A − B ( x ) ≥ τ }| ≪ | A || B | τ d ( A )was established, where d ( A ) = min C = ∅ | AC | | A || C | . See [8, Lemma 7] for a proof. As pointed outin [3], d ∗ ( A ) ≤ d ( A ), since for any non empty C we can take t = | C | , Q = AC and R = C − in the definition of d ∗ ( A ).3. A bound on sums of multiplicative energies with shifts
The aim of this section is to prove the first of the two main new lemmas of this paper.This is the following lemma, which gives an improvement of Lemma 2.4 in [7], in the casewhen the sets involved are approximately the same size, unless the multiplicative energy isessentially as large as possible.
Lemma 8.
Let
A, B, C ⊂ R be a finite sets such that | A | ≈ | C | . Then X a ∈ A E × ( B, C − a ) ≪ E × ( B ) / | A | log / | A | + | A | + | A || B | . B. MURPHY, O. ROCHE-NEWTON AND I. SHKREDOV
Proof.
We have X a ∈ A E × ( B, C − a ) = |{ ( a, b, b ′ , c, c ′ ) ∈ A × B × B × C × C : b ( c − a ) = b ′ ( c ′ − a ) }|≤ |{ ( a, b, b ′ , c, c ′ ) ∈ A × B × B × C × C : b ( c − a ) = b ′ ( c ′ − a ) = 0 }| + | A | + | A | | B | + | A || B | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( a, b, b ′ , c, c ′ ) ∈ A × B × B × C × C, : bb ′ = c ′ − ac − a = 0 (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + | A | + E × ( B ) / | A | + | A || B | . The remaining task is to bound the main term. Applying the Cauchy-Schwarz inequalityyields (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( a, b, b ′ , c, c ′ ) ∈ A × B × B × C × C : bb ′ = c ′ − ac − a = 0 (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = X x =0 r B/B ( x ) n ( x ) ≤ X x r B/B ( x ) ! / X x =0 n ( x ) ! / = E × ( B ) / X x =0 n ( x ) ! / , (3.1)where n ( x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( a, c, c ′ ) ∈ A × C × C : x = c ′ − ac − a (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) . Note that X x n ( x ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( a , a , c , c , c ′ , c ′ ) : c ′ − a c − a = c ′ − a c − a (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( a , a , c , c , c ′ , c ′ ) : c − a c − a = c ′ − a c ′ − a (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) . The identity c − a c − a = c ′ − a c ′ − a occurs only if the three points ( a , a ) , ( c , c ) , ( c ′ , c ′ ) ∈ ( A ∪ C ) × ( A ∪ C ) are collinear. By Lemma 6, there are O ( | A ∪ C | log | A | ) such collinear triples, andso X x n ( x ) ≪ | A | log | A | . Combining this with (3.1), we have X a ∈ A E × ( B, C − a ) ≪ E × ( B ) / | A | log / | A | + | A | + | A || B | . ARIATIONS ON THE SUM-PRODUCT PROBLEM II 7 (cid:3)
Corollary 9.
Let A ⊂ R be finite. Then there exists a ∈ A such that (3.2) E × ( A ) | A ( A + a ) | ≫ | A | log | A | . Similarly, there exists b ∈ A such that (3.3) E × ( A ) | A ( A − b ) | ≫ | A | log | A | . Proof.
By Lemma 8, we have X a ∈ A E × ( A, A + a ) ≪ E × ( A ) / | A | log / | A | + | A | ≪ E × ( A ) / | A | log / | A | . Therefore, by the pigeonhole principle, there exists a ∈ A such that E × ( A, A + a ) ≪ E × ( A ) / | A | log / | A | . By the Cauchy-Schwarz inequality | A | | A ( A + a ) | ≤ E × ( A, A + a ) ≪ E × ( A ) / | A | log / | A | . A rearrangement of this inequality completes the proof. The proof of (3.3) is essentiallyidentical. (cid:3)
Corollary 9 provides a pinned version of the following result, which was the main lemmafrom [7].
Lemma 10.
For any finite sets
A, B, C ∈ R , E × ( A ) | A ( B + C ) | ≫ | A | | B || C | log | A | . A conditional lower bound on | A ( A + α ) | The second main new lemma of this paper is the following.
Lemma 11.
Let A ⊂ R and α ∈ R \ { } . Then | A ( A + α ) | & E × ( A ) | A | d ∗ ( A ) . In particular, this result implies the following statement: | AA | ≤ M | A | ⇒ | A ( A + α ) | ≫ M | A | + c . B. MURPHY, O. ROCHE-NEWTON AND I. SHKREDOV
Indeed, since by the Cauchy-Schwarz inequality E × ( A ) ≥ | A | /M and since d ∗ ( A ) ≤ d ( A ) ≤ M , we have | A ( A + α ) | & | A | M . In the proof of Lemma 11 we will need the following simple lemma. See [14, Lemma 4].
Lemma 12.
Let G be an abelian group and A ⊂ G be a finite set. Then there is z such that X x ∈ zA | ( zA ) ∩ x ( zA ) | ≫ E × ( A ) | A | log | A | . We will also need the following result of the third author [13], which tells us that a set A such that the parameter d ∗ ( A ) is small has small additive energy. One can obtain similarbut weaker results using the Szemer´edi-Trotter Theorem in a more elementary way, see forexample Corollary 8 in [8], but the important thing for our application of this result is thatthe exponent 32 /
13 is smaller than 5 / Theorem 13 ([12], Theorem 5.4) . Let A ⊂ R be a finite set. Then E + ( A ) ≪ d ∗ ( A ) / | A | / log / | A | . The result in [13] did not use the quantity d ∗ ( A ) because it was introduced later in [4] butone can check that the arguments of [13] work for any Szemer´edi-Trotter set. The only factwe need is an upper bound for size of a set { x : r A − B ( x ) ≥ τ } , so Lemma 7 is enough forus. Proof of Lemma 11.
Without loss of generality, we may assume that 0 / ∈ A . ApplyingLemma 12 and writing B = zA , we have(4.1) X λ ∈ B | B ∩ λ − B | ≫ E × ( A ) | A | log | A | . Next we double count the number of solution to the equation(4.2) b ( b ′ + αz ) = b ( b ′ + αz )such that b , b ∈ B , b ′ ∈ B b and b ′ ∈ B b .Let S denote the number of solutions to (4.2). Suppose that we have such a solution.Then b ′ = b ′′ /b and b ′ = b ′′ /b for some b ′′ , b ′′ ∈ B . Therefore αzb + b ′′ = αzb + b ′′ , and it follows that S ≤ E + ( B, αzB ). An application of the Cauchy-Schwarz inequality thengives E + ( B, αzB ) ≤ E + ( B ) / E + ( αzB ) / = E + ( B ) = E + ( A ). So, by Theorem 13(4.3) S ≤ E + ( A ) . d ∗ ( A ) / | A | / . ARIATIONS ON THE SUM-PRODUCT PROBLEM II 9
On the other hand denote n ( t ) = |{ ( b, b ′ ) ∈ B × B b : b ( b ′ + αz ) = t }| and note that S = P t n ( t ). Also, n ( t ) > t ∈ B ( B + αz ). Then, by (4.1), theCauchy-Schwarz inequality and (4.3) E × ( A ) | A | log | A | ≪ X λ ∈ B | B ∩ λ − B | ! (4.4) = X t n ( t ) ! (4.5) ≤ | B ( B + αz ) | X t n ( t )(4.6) . | B ( B + αz ) | d ∗ ( A ) / | A | / . (4.7)Finally, note that | B ( B + αz ) | = | A ( A + α ) | . We conclude that | A ( A + α ) | & E × ( A ) | A | d ∗ ( A ) . (cid:3) Actually, one can see that we have proved the inequality | A | | A ( A + α ) | E + ( A ) & ( E × ( A )) for any finite subset of reals. 5. Proof of Theorem 1
Before proving Theorem 1, we need one more lemma.
Lemma 14.
Let A ⊂ R and suppose that E × ( A ) ≥ | A | K . Then for any α ∈ R \ { }| A ( A + α ) | & | A | / K / . Proof.
We claim that for any A ⊂ R such that E × ( A ) ≥ | A | K there is a subset A ′ ⊆ A suchthat E × ( A ′ ) & E × ( A ) ≥ | A | K , (5.1) d ∗ ( A ′ ) . K | A || A ′ | . (5.2) Given such a subset A ′ , we may apply Lemma 11 to find that | A ( A + α ) | ≥ | A ′ ( A ′ + α ) | & E × ( A ′ ) | A ′ | / d / ∗ ( A ′ ) & | A | K | A ′ | / · | A ′ | / | A | / K / = | A | / K / | A ′ | / ≥ | A | / K / . It remains to prove (5.1) and (5.2). By the popularity principle and dyadic pigeonholingthere is a subset P ⊆ A/A and a number ∆ ≥ | A | / K such that for all x in P ∆ ≤ | A ∩ xA | < , and X x ∈ P | A ∩ xA | & E × ( A ) . Now we perform an additional refinement step. Let A ′ ⊆ A denote the set of x such that | P ∩ xA − | ≥ ∆ | P | | A | . Since X x ∈ A | P ∩ xA − | = X x ∈ P | A ∩ xA | ≥ ∆ | P | , by the popularity principle we have X x ∈ A ′ | P ∩ xA − | ≥ | P | . If x A ′ , then | P ∩ x ( A ′ ) − | ≤ | P ∩ xA − | < ∆ | P | | A | . Thus 3∆ | P | ≤ X x ∈ A ′ | P ∩ xA − | = X x ∈ A | P ∩ x ( A ′ ) − | ≤ ∆ | P | X x ∈ A ′ | P ∩ x ( A ′ ) − | , which yields ∆ | P | ≤ X x ∈ A ′ | P ∩ x ( A ′ ) − | = X x ∈ P | A ′ ∩ xA ′ | . By Cauchy-Schwarz, we have E × ( A ′ ) ≫ ∆ | P | & E × ( A ) . ARIATIONS ON THE SUM-PRODUCT PROBLEM II 11
Setting Q = P , R = A , and t = (∆ | P | ) / (4 | A | ) in the definition of the quantity d ∗ ( A ′ ), weobtain d ∗ ( A ′ ) ≪ | P | | A | (cid:16) ∆ | P | | A | (cid:17) | A ′ | ≪ | A | | A ′ || P | ∆ . | A | | A ′ | E × ( A )∆ ≪ K | A || A ′ | , where the last inequality follows from the lower bounds for E × ( A ) and ∆. (cid:3) Proof of Theorem 1.
Write E × ( A ) = | A | K . By Corollary 9 there is some a ∈ A such that(5.3) | A ( A + a ) | ≫ K / | A | / log − / | A | & K / | A | / . On the other hand, for any a ∈ A \ { } , Lemma 14 implies that(5.4) | A ( A + a ) | & | A | / K / . Optimizing over (5.3) and (5.4), we obtain | A ( A + a ) | & | A | / / as required. (cid:3) In fact, by taking more care with the pigeonholing argument in the proof of Corollary 9it follows that the bound | A ( A + a ) | & | A | / / holds for at least half of the elements a ∈ A . 6. Three variable expanders
The proofs of Theorems 2 and 3 follow a similar argument to that of Theorem 1. Wecan use Corollary 9 to get an exponent better than 3 / E × ( A ) is not toolarge. However, in the case when E × ( A ) is large, we need analogues of Lemma 14 that arequantitatively better for the purposes of these problems. These bounds are given by thefollowing lemma. Lemma 15.
Let A ⊂ R and suppose that E × ( A ) ≥ | A | K . Then (6.1) | A − A | & | A | / K / and (6.2) | A + A | & | A | / K / . In order to prove (6.1), we will need the following lemma.
Lemma 16.
For any finite set A ⊂ R , | A − A | ≫ | A | / d / ∗ ( A ) log / | A | . Although this result has not appeared explicitly in the literature, it can be proved byessentially copying the arguments from [8] and predecessors with the stronger Lemma 7 inplace of the bound (2.2). The proof is included in the appendix for completeness.The following similar result for sum sets follows from a combination of the work in [12]and [3]:
Lemma 17.
For any finite set A ⊂ R , | A + A | & | A | / d / ∗ ( A ) . To be more precise, it was proven in [12] that if A is a Szemer´edi-Trotter set with parameter D ( A ), then | A + A | & | A | / D / ( A ) , and it was subsequently established in [3] that any set A isa Szemer´edi-Trotter set with O ( d ∗ ( A )).In addition, we need the following lemma, which uses the hypothesis that the energy islarge in order to find a large subset A ′ ⊂ A such that d ∗ ( A ) is small. Lemma 18 (Double pigeonholing argument) . Let A ⊂ R and suppose that E × ( A ) ≥ | A | K .Then there is a subset A ′ ⊆ A and a number ∆ ≫ | A | /K such that | A ′ | & | A | K ∆(6.3) d ∗ ( A ′ ) . K | A ′ | | A | ∆ . (6.4)The proof of Lemma 18 is similar to the refinement step in the proof of Lemma 14. Proof.
By the popularity principle and dyadic pigeonholing there is a subset P ⊆ A/A anda number ∆ ≥ | A | / K such that for all x in P ∆ ≤ | A ∩ xA | < , and X x ∈ P | A ∩ xA | & E × ( A ) . Now we perform a second dyadic pigeonholing argument. Note that X a ∈ A | A ∩ aP | = X x ∈ P | A ∩ xA | ≥ | P | ∆ . Thus there exists a subset A ′ ⊆ A and a number 0 < t ≤ | A | such that for all a in A ′ t ≤ | A ∩ aP | < t and X a ∈ A ′ | A ∩ aP | & | P | ∆ , ARIATIONS ON THE SUM-PRODUCT PROBLEM II 13 hence | A ′ | t & | P | ∆ . Since t ≤ | A | and | P | ∆ & | A | /K we have | A ′ | & | P | ∆ | A | & | A | K ∆ , which proves (6.3).For every a ∈ A ′ we have | A ∩ aP | ≥ t . Therefore we can take t = t, Q = A, and R = P − in the definition of d ∗ ( A ′ ). We then have d ∗ ( A ′ ) ≤ | A | | P | | A ′ | t . | A | | P | | A ′ | t · (cid:18) | A ′ | t | P | ∆ (cid:19) = | A | | A ′ | | P | ∆ = | A | | A ′ | ( | P | ∆ )∆ . K | A ′ | | A | ∆ , which proves (6.4). (cid:3) Proof of Lemma 15.
Similar to the proof of Lemma 14, the idea here is to use the doublepigeonholing argument (Lemma 18) to find a large subset A ′ ⊂ A such that d ∗ ( A ) is small,and to then apply Lemmas 16 and 17 to complete the proof.Since E × ( A ) ≥ | A | /K , by Lemma 18 there is a subset A ′ ⊆ A and a number ∆ ≫ | A | /K such that | A ′ | & | A | K ∆ d ∗ ( A ′ ) . K | A ′ | | A | ∆ . Applying Lemma 16 yields | A − A | ≥ | A ′ − A ′ | & | A ′ | / d ∗ ( A ′ ) / & | A ′ | / · | A | / ∆ / | A ′ | / K / = | A ′ | / | A | / ∆ / K / & | A | / K / ∆ / · | A | / ∆ / K / = | A | / K ∆ / ≫ | A | / K / . Applying Lemma 17 yields | A + A | ≥ | A ′ + A ′ | & | A ′ | / d / ∗ ( A ′ ) & | A ′ | / · | A | / ∆ / | A ′ | / K / = | A ′ | / | A | / ∆ / K / & | A | / K / ∆ / · | A | / ∆ / K / = | A | / K ∆ / ≫ | A | / K / . (cid:3) We are now ready to prove the new lower bounds for A ( A − A ) and A ( A + A ). Proof of Theorem 2.
Write E × ( A ) = | A | K . By Corollary 9(6.5) | A ( A + A ) | ≫ K / | A | / log − / | A | & K / | A | / . On the other hand, Lemma 15 implies that(6.6) | A ( A + A ) | ≥ | A + A | & | A | / K / Optimizing over (6.5) and (6.6), we obtain | A ( A + A ) | & | A | / / as required. (cid:3) Proof of Theorem 3.
Write E × ( A ) = | A | K . By Corollary 9(6.7) | A ( A − A ) | ≫ K / | A | / log − / | A | & K / | A | / . On the other hand, Lemma 15 implies that(6.8) | A ( A − A ) | ≥ | A − A | & | A | / K / Optimizing over (6.7) and (6.8), we obtain | A ( A − A ) | & | A | / / as required. (cid:3) ARIATIONS ON THE SUM-PRODUCT PROBLEM II 15 Five variable expander
In this section we will prove Theorem 4, based on the proof in [7] of the inequality(7.1) | A ( A + A + A + A ) | ≫ | A | log | A | . The proof of (7.1) in [7] follows from comparing the upper bound on the multiplicativeenergy in Theorem 5 with the lower bound in Lemma 10. Here, we need a suitable analogueof Lemma 10, the proof of which relies on the following celebrated result of Guth and Katz[2].
Theorem 19.
For any finite set A ⊂ R , the number of solutions to the equation ( a − a ) + ( a − a ) = ( a − a ) + ( a − a ) such that a , . . . , a ∈ A is at most O ( | A | log | A | ) . Theorem 19 is a special case of a more general geometric result which immediately impliesa resolution of the Erd˝os distinct distances problem up to logarithmic factors, but here it isstated only in the form in which it will be used in this paper.Theorem 19 can be used to prove the following variation of Lemma 10:
Lemma 20.
For any finite sets
A, B ⊂ R , E + ( A ) |{ a + ( b + b ) : a ∈ A, b , b ∈ B }| ≫ | A | | B | log | B | . Proof.
The proof proceeds by the familiar method of double counting the number of solutionsto the equation(7.2) a + ( b + b ) = a + ( b + b ) such that a i ∈ A and b i ∈ B . Let S denote the number of solutions to (7.2) and write A + ( B + B ) := { a + ( b + b ) : a ∈ A, b , b ∈ B } . By the Cauchy-Schwarz inequality S ≥ | A | | B | | A + ( B + B ) | . On the other hand, also by the Cauchy-Schwarz inequality, S = X x r A − A ( x ) r ( B + B ) − ( B + B ) ( x ) ! ≤ X x r A − A ( x ) ! X x r B + B ) − ( B + B ) ( x ) ! = E + ( A ) X x r B + B ) − ( B + B ) ( x ) ! . Theorem 19 tells us that (cid:16)P x r B + B ) − ( B + B ) ( x ) (cid:17) = O ( | B | log | B | ). Therefore, | A | | B | ≤ | A + ( B + B ) | S ≪ | A + ( B + B ) | E + ( A ) | B | log | B | . After rearranging this inequality, we obtain the desired result. (cid:3)
Unfortunately, we are not aware of a proof of Lemma 20 which does not use the deepresults from [2].We are now ready to prove Theorem 4.
Proof of Theorem 4.
Apply Lemma 20 with A = log A and B = A + A . We have E + (log A ) |{ ( a + a + a + a ) + log a : a i ∈ A }| ≫ | A | | A + A | log | A | . Note that log a + log a = log a + log a if and only if a a = a a , and so E + (log A ) = E × ( A ). We can apply Theorem 5 to deduce that E + (log A ) ≪ | A + A | log | A | . It then follows that |{ ( a + a + a + a ) + log a : a i ∈ A }| ≫ | A | log | A | , which completes the proof. (cid:3) Appendix
The purpose of this appendix is to present a formal proof of Lemma 16. We will call uponthe following result on the relationship between different types of energy.
Lemma 21 ([5], Lemma 2.4 and 2.5) . For any finite sets
A, B ⊂ R | A | ( E +1 . ( A )) ≤ ( E +3 ( A )) / ( E +3 ( B )) / E ( A, A − B ) . In fact, Lemma 21 holds for any abelian group.
Proof of Lemma 16.
Recall that Lemma 16 states that for any finite set A ⊂ R , | A − A | ≫ | A | / d / ∗ ( A ) log / | A | . ARIATIONS ON THE SUM-PRODUCT PROBLEM II 17
In order to prove this, we will first prove two energy bounds. Note that, by Lemma 7, E +3 ( A ) = X x r A − A ( x )= X j ≥ X x :2 j − ≤ r A − A ( x ) < j r A − A ( x ) ≪ | A | d ∗ ( A ) log | A | . (7.3)Similarly, for any F ⊂ R , and a parameter △ > E + ( A, F ) = X x r A − F ( x )= X x : r A − F ( x ) ≤△ r A − F ( x ) + X j ≥ X x :2 j − △≤ r A − A ( x ) < j △ r A − F ( x ) ≪ △| A || F | + | A || F | d ∗ ( A ) △ . (7.4)We choose △ = ( | F | d ∗ ( A )) / , and thus conclude that(7.5) E ( A, F ) ≪ | A || F | / d ∗ ( A ) / . Now, by H¨older’s inequality | A | = X x ∈ A − A r A − A ( x ) ! ≤ | A − A | X x r / A − A ( x ) ! = | A − A | ( E +1 . ( A )) . (7.6)Applying (7.6) with Lemma 21, as well as inequalities (7.3) and (7.5), we have | A | ≤ | A − A | ( E +1 . ( A )) | A | ≤ | A − A | E +3 ( A ) E ( A, A − A )= | A − A | / | A | d ∗ ( A ) / log | A | . Rearranging this inequality completes the proof. (cid:3)
Finally, we note that a similar method can be used to prove a quantitatively weaker versionof Lemma 17, in the form of the following result:(7.7) | A + A | ≫ | A | / d / ∗ ( A ) log / | A | . To see how this works, one can repeat the arguments from the proof of Theorem 1.2 in [6],but using Lemma 7 in place of Lemma 3.2 from [6]. This is worth noting, since the proofsof the main results in [3] and [4], that is the boundmax {| A + A | , | AA |} ≫ | A | / c for some c >
0, both include applications of Lemma 17. One can also obtain this sum-product estimate, albeit with a smaller positive value c , by using the bound (7.7) instead ofLemma 17. Acknowledgement
The second author is grateful for the hospitality of the R´enyi Institute, where part of thiswork was conducted with support from Grant ERC-AdG. 321104, and also for the supportof the Austrian Science Fund FWF Project F5511-N26, which is part of the Special ResearchProgram ”Quasi-Monte Carlo Methods: Theory and Applications”. We thank Antal Balog,Orit Raz and Endre Szemer´edi for helpful discussions.
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ARIATIONS ON THE SUM-PRODUCT PROBLEM II 19
B. Murphy: School of Mathematics, University Walk, Bristol, UK, BS8 1TW, and Heil-bronn Institute of Mathematical Research
E-mail address : [email protected] O. Roche-Newton: 69 Altenberger Straße, Johannes Kepler Universit¨at, Linz, Austria
E-mail address : [email protected] I. D. Shkredov: Steklov Mathematical Institute, ul. Gubkina, 8, Moscow, Russia, 119991,IITP RAS, Bolshoy Karetny per. 19, Moscow, Russia, 127994, and MIPT, Institutskii per.9, Dolgoprudnii, Russia, 141701
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