aa r X i v : . [ m a t h . C O ] A ug ON A THEOREM OF SCHOEN AND SHKREDOV ON SUMSETS OFCONVEX SETS
LIANGPAN LI
Abstract.
A set of reals A = { a , . . . , a n } labeled in increasing order is called convexif there exists a continuous strictly convex function f such that f ( i ) = a i for every i .Given a convex set A , we prove | A + A | ≫ | A | / (log | A | ) / . Sumsets of different summands and an application to a sum-product-type problem arealso studied either as remarks or as theorems. Introduction
Let A = { a , . . . , a n } be a set of real numbers labeled in increasing order. We say that A is convex if there exists a continuous strictly convex function f such that f ( i ) = a i forevery i . Hegyv´ari ([10]), confirming a conjecture of Erd˝os, proved that if A is convex then | A − A | ≫ | A | · log | A | log log | A | , where “ ≫ ” is the Vinogradov notation. This result was later improved by many authors,see for example [5, 8, 9, 11, 14, 23] for related results. Recently, Schoen and Shkredov([21]), combining an energy-type equality ([20]) E ( A ) = X s E (cid:0) A, A ∩ ( A + s ) (cid:1) , (1.1)a useful set inclusion relation (see e.g. [13, 17, 18, 19, 20]) | ( A + A ) ∩ ( A + A + s ) | ≥ | A + ( A ∩ ( A + s )) | , (1.2)and an application (see Lemma 2.1 below) of the Szemer´edi-Trotter incidence theorem(see e.g. [12, 24, 25]), proved for convex sets the following best currently known lowerbounds: | A + A | ≫ | A | / (log | A | ) / , (1.3) | A − A | ≫ | A | / (log | A | ) / . (1.4)We also remark that Solymosi and Szemer´edi obtained a similar result for convex sets,establishing | A ± A | ≫ | A | . δ for some universal constant δ > Mathematics Subject Classification.
Key words and phrases. sumset, productset, convex set, energy, Szemer´edi-Trotter theorem.
The purpose of this note is twofold. Firstly, we give a slight improvement of (1.3) asfollows:
Theorem 1.1.
Let A be a convex set. Then | A + A | ≫ | A | / (log | A | ) / . (1.5)Secondly, and most importantly, we will address an application of the Schoen-Shkredovestimate to a sum-product-type problem. Erd˝os and Szemer´edi ([7]) once conjectured thatthe size of either the sumset or the productset of an arbitrary set of the reals must bevery large, see [22] for the best currently known result toward this conjecture and relatedreferences therein. Another type of problem than one can attack regarding sumset andproductset is to assume either one is very small, then prove the other one is very large.Elekes and Ruzsa ([6], see also [16, 22]) proved that if the sumset of a set is very small,then its productset must be very large. On the other hand, if the productset of a set isvery small, say for example | AA | ≤ M | A | , then the best currently known lower bound forthe size of its sumset ([4], see also [5, 16, 22]) only is | A + A | ≥ C M | A | / .Roughly speaking, we will show that a set with very small multiplicative doubling is a“convex” set. Consequently, we can derive the following improvement. Theorem 1.2.
Suppose | AA | ≤ M | A | . Then | A + A | ≫ M | A | / (log | A | ) / , | A − A | ≫ M | A | / (log | A | ) / . We remark that one can find direct application of Theorem 1.2 to the main result in [15],in which multi-fold sums from a set with very small multiplicative doubling are studied.See also [1, 2, 3] for some related discussions on multi-fold sumsets.We collect some notations used throughout this note. Denote by δ A,B ( s ) the number ofrepresentations of s in the form a − b , a ∈ A , b ∈ B . If A = B we write δ A ( s ) = δ A,A ( s )for simplicity. Furthermore, put E ( A, B ) = X s δ A ( s ) δ B ( s ) = X s δ A,B ( s ) and E k ( A ) = X s δ A ( s ) k . Let A s = A ∩ ( A + s ). All logarithms are to base 2. All sets are finite subsets of real numbers .2.
Convexity and energy estimates
Lemma 2.1 ([21]) . Let A be a convex set. Then for any set B and any τ ≥ we have (cid:12)(cid:12) { x ∈ A − B : δ A,B ( x ) ≥ τ } (cid:12)(cid:12) ≪ | A | · | B | τ . A special case of Lemma 2.1 for B = − A was established in [11]. As applications, wehave the following two lemmas. UMSETS OF CONVEX SETS 3
Lemma 2.2 ([21]) . Let A be a convex set. Then E ( A ) ≪ | A | · log | A | . Lemma 2.3.
Let A be a convex set. Then for any set B we have E ( A, B ) ≪ | A | · | B | . .Proof. Let △ . = E ( A,B )2 | A || B | and we divide E ( A, B ) into two parts, one is X s : δ A,B ( s ) < △ δ A,B ( s ) , which is obviously less than half of E ( A, B ), thus results in the other part X s : δ A,B ( s ) ≥△ δ A,B ( s ) , being bigger than half of E ( A, B ). Therefore, by Lemma 2.1 and a dyadic argument, E ( A, B )2 ≤ X s : δ A,B ( s ) ≥△ δ A,B ( s ) ≪ X j ≥ △ · j · | A | · | B | △ · j ≤ | A | · | B | △ . This finishes the proof. (cid:3)
Lemma 2.4.
Let
A, B be any sets. Then X s E ( A s , B ) ≤ E ( A ) / · E ( B ) / . Proof.
Note δ A s ( t ) = δ A t ( s ), which in common is | A ∩ ( A + s ) ∩ ( A + t ) ∩ ( A + s + t ) | . Thus X s E ( A s , B ) = X s X t δ A s ( t ) δ B ( t ) = X s X t δ A t ( s ) δ B ( t )= X t X s δ A t ( s ) δ B ( t ) = X t δ A ( t ) δ B ( t ) ≤ (cid:16) X t δ A ( t ) (cid:17) / · (cid:16) X t δ B ( t ) (cid:17) / = E ( A ) / · E ( B ) / . This finishes the proof. (cid:3)
Lemma 2.5.
Let
A, B be any sets. Then E . ( A ) · | B | ≤ (cid:0) X s E ( A s , B ) (cid:1) · E ( A, A + B ) . Proof.
By the Cauchy-Schwarz inequality, | A s | . · | B | ≤ E ( A s , B ) / · | A s + B | / · | A s | / . First summing over all s ∈ A − A , then applying Cauchy-Schwarz again gives E . ( A ) · | B | ≤ (cid:0) X s E ( A s , B ) (cid:1) · (cid:0) X s | A s + B | · | A s | (cid:1) ≤ (cid:0) X s E ( A s , B ) (cid:1) · (cid:0) X s | ( A + B ) s | · | A s | (cid:1) = (cid:0) X s E ( A s , B ) (cid:1) · E ( A, A + B ) , where the second inequality is due to the set inclusion relation A s + B ⊂ ( A + B ) s . Thisfinishes the proof. (cid:3) LIANGPAN LI Proof of Theorem 1.1
This section is mainly devoted to the proof of Theorem 1.1. We first claim E ( A ) ≪ | A | · E . ( A ) , which follows simply from (see also the proof of Lemma 2.3) E ( A ) = X s : δ A ( s ) < △ δ A ( s ) + X s : δ A ( s ) ≥△ δ A ( s ) ≪ p △ · E . ( A ) + | A | △ . Then applying Lemma 2.5 with B = A , Lemma 2.4 and Lemma 2.2, we get | A | | A + A | ≤ E ( A ) ≪ | A | · | A | · (log | A | ) · | A | · | A + A | / , which is equivalent to | A + A | ≫ | A | / (log | A | ) / . This finishes the proof of Theorem 1.1.
Remark 3.1.
Let
A, B be convex sets. We remark that one can establish | A ± B | ≫ | A | · | B | (log | A | ) / · (log | B | ) / . (3.1)To this aim, it suffices to note | A | · | B | | A ± B | ≤ E ( A, B ) = X s δ A ( s ) · δ B ( s ) ≤ (cid:0) X s δ A ( s ) / (cid:1) / · (cid:0) X s δ B ( s ) (cid:1) / = E . ( A ) / · E ( B ) / , then turning to Lemmas 2.2 ∼ Remark 3.2.
Let
A, B be convex sets. We remark that one can establish | A − A | · | A ± B | ≫ | A | · | B | (log | A | ) / · (log | B | ) / . (3.2)To this aim, it suffices to note from the H¨older inequality that | A | | A − A | ≤ E . ( A ) , then turning to Lemmas 2.2 ∼ Proof of Theorem 1.2
Lemma 4.1.
Let A be a set of the form f ( Z ) , where f is a continuous strictly convexfunction, | Z + Z | ≤ M | Z | . Then for any set B and any τ ≥ , (cid:12)(cid:12) { x ∈ A − B : δ A,B ( x ) ≥ τ } ≪ M · | A | · | B | τ . UMSETS OF CONVEX SETS 5
Proof.
Without loss of generality, we may assume that f is monotonically increasing, and1 ≪ τ ≤ min {| A | , | B |} . Let G ( f ) denote the graph of f in the plane. For any ( α, β ) ∈ R ,put L α,β = G ( f ) + ( α, − β ) . Define the pseudo-line system L = { L z,b : ( z, b ) ∈ Z × B } , andthe set of points P = ( Z + Z ) × ( A − B ) . By convexity, |L| = | Z | · | B | = | A | · | B | . Let P τ be the set of points of P belonging to at least τ curves from L . By the Szemer´edi-Trotterincidence theorem, τ · |P τ | ≪ ( |P τ | · | Z | · | B | ) / + | Z | · | B | + |P τ | , from which we can deduce (see also [21]) |P τ | ≪ | Z | · | B | τ . Next, suppose δ A,B ( x ) ≥ τ . There exist τ distinct elements { z i } τi =1 from Z , τ distinctelements { b i } τi =1 from B , such that x = f ( z i ) − b i ( ∀ i ). Now we define Z i , z i + Z ( ∀ i )and M x ( s ) , P τi =1 χ Z i ( s ), where χ Z i ( · ) is the characteristic function of Z i . Since( z i + z, x ) = (cid:0) z i , f ( z i ) (cid:1) + ( z, − b i ) ∈ L z,b i ( ∀ z, ∀ i ) , we have ( s, x ) ∈ P M x ( s ) . Obviously, X s ∈ Z + Z M x ( s ) = τ X i =1 X s ∈ Z + Z χ Z i ( s ) ≥ τ | Z | . Thus by the standard popularity argument, (cid:12)(cid:12) { s ∈ Z + Z : M x ( s ) ≥ τ M } (cid:12)(cid:12) ≥ | Z | . This naturally implies (cid:12)(cid:12) { x ∈ A − B : δ A,B ( x ) ≥ τ } (cid:12)(cid:12) · | Z | ≤ |P τ M | , and consequently, (cid:12)(cid:12) { x ∈ A − B : δ A,B ( x ) ≥ τ } (cid:12)(cid:12) ≪ |P τ M || Z | ≪ M · | Z | · | B | τ = M · | A | · | B | τ . This finishes the proof. (cid:3)
It is rather easy to observe that, any property holds for convex sets in this note shouldalso hold for sets of the form f ( Z ), where f is a continuous strictly convex function, | Z + Z | ≤ M | Z | , with ≫ replaced by ≫ M .As applications, let A be a finite set of positive real numbers with | AA | ≤ M | A | . Then A = exp( Z ), Z = ln A , | Z + Z | = | AA | ≤ M | A | = M | Z | . Consequently, (1.5) and (3.2)hold for such an A . This suffices to prove Theorem 1.2. We are done. Acknowledgements.
This work was supported by the NSF of China (11001174).
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