Konyagin, S. V., & Rudnev, M. (2013). On new sum-product-type estimates. SIAM Journal on Discrete Mathematics, 27(2), DOI: 10.

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1 Konyagin, S. V., & Rudnev, M. (013). On new sum-produc-ype esimaes. SIAM Journal on Discree Mahemaics, 7(), DOI: / Publisher's PDF, also known as Version of record Link o published version (if available): / Link o publicaion record in Explore Brisol Research PDF-documen This is he final published version of he aricle (version of record). I firs appeared online via Sociey for Indusrial and Applied Mahemaics a hp://dx.doi.org/ / Please refer o any applicable erms of use of he publisher. Universiy of Brisol - Explore Brisol Research General righs This documen is made available in accordance wih publisher policies. Please cie only he published version using he reference above. Full erms of use are available: hp://

2 SIAM J. DISCRETE MATH. Vol. 7, No., pp c 013 Sociey for Indusrial and Applied Mahemaics ON NEW SUM-PRODUCT TYPE ESTIMATES SERGEI V. KONYAGIN AND MISHA RUDNEV Absrac. New lower bounds involving sum, difference, produc, and raio ses of a se A C are given. The esimaes involving he sum se mach, up o consans, he sae-of-he-ar esimaes, proven by Solymosi for he reals and are obained by generalizing his approach o he complex plane. The bounds involving he difference se improve he currenly bes known ones, also due o Solymosi, in boh he real and complex cases by means of combining he Szemerédi Troer heorem wih an arihmeic combinaorics echnique. Key words. Erdős Szemerédi conjecure, sum-produc esimaes, Szemerédi Troer heorem, incidence heory AMS subjec classificaions. 68R05, 11B75 DOI / Inroducion. Erdős and Szemerédi [4] conjecured ha if A is a finie se of inegers, hen for any ε>0, as he cardinaliy A, Above, A + A + A A A ε. A + A = {a 1 + a : a 1,a A} is called he sum se of A, he produc A A, difference A A, andraioa : A ses being similarly defined. (In he laer case one should no divide by zero.) Variaions of he Erdős Szemerédi conjecure address subses of oher rings or fields see [18] for a general discussion and [1] for a new quaniaive sum-produc esimae in funcion fields as well as replacing, e.g., he sum se wih he difference se A A. The conjecure is far from being seled, and herefore curren world records vary wih such variaions of he problem. The bes resul for A R, for insance, is due o Solymosi [16], claiming (1.1) A + A + A A A log 1 3 A, and wihou he logarihmic erm if A A is replaced by A : A. The noaion, is being used hroughou o suppress absolue consans in inequaliies, ha is, consans which do no depend on he parameer A. A firs glance, he consrucion in [16] appears o be specific for reals, and i does no seem o allow for replacing he sum se A + A wih he difference se A A. So, Received by he ediors July 30, 01; acceped for publicaion (in revised form) March 6, 013; published elecronically May, 013. hp:// Seklov Mahemaical Insiue, Moscow , Russia (konyagin@mi.ras.ru). This auhor s work was parially suppored by he Russian Fund for Basic Research, gran , and by he Program Supporing Leading Scienific Schools, gran Nsh Deparmen of Mahemaics, Universiy of Brisol, Brisol BS8 1TW, Unied Kingdom (m.rudnev@brisol.ac.uk). 973

3 974 SERGEI V. KONYAGIN AND MISHA RUDNEV if A C or if A + A for reals ges replaced by A A, he bes known resul comes from an older paper of Solymosi [15], claiming (1.) A A + A A A , log 3 11 A and wihou he logarihmic erm if A A ges replaced by A : A. In his paper we show, firs, ha he order-based observaion which allowed Solymosi o prove (1.1), namely he fac ha for real posiive a, b, c, d ( a b d) < c ( a b < a + c b + d < c ), d admis a naural exension o he complex case. We herefore exend he esimae (1.1) o he case A C. This is he conen of he forhcoming Theorem 1.1. Second, we prove new esimaes involving he difference se, for A C, which improve on (1.). For his we use raher differen argumens, relying on he Szemerédi Troer heorem combined wih an arihmeic echnique. This is he conen of he forhcoming Theorem 1.. We remark ha Theorems 1.1 and 1., even hough hey apply o he case A C, boh rely crucially on he meric properies of he Euclidean space, and we presenly do no see how he ideas behind hem could apply o he case when A is a small subse of a prime residue field Z p of large characerisic, where he bes known exponen in he sum-produc inequaliy is 1 11, up o a logarihmic facor in A ; see [10]. We now formulae our main resuls. Theorem 1.1. For any finie A C wih a leas wo elemens, one has he following esimaes: (1.3) Theorem 1.. following esimaes: (1.4) A + A + A : A A , A + A + A A A log 1 3 A. For any finie A C wih a leas wo elemens, one has he A A + A : A A , log 4 31 A 11 A A A + A A. log 5 13 A. Preliminary seup. In his secion we develop he preliminary seup and noaion o be used in he forhcoming proofs of Theorems 1.1 and 1.. Since we do no pursue bes possible values of he consans, hidden in he inequaliies (1.3), (1.4), we furher assume ha 0 A and A C for some absolue consan C, which is as large as necessary. Observe ha Theorems 1.1 and 1. each claim wo differen esimaes: one involving he raio se A : A and he oher involving he produc se A A. In order o prove hese esimaes, we deal wih a cerain popular subse P of he poin se A A C.Noehaifl A: A is a raio, i can be idenified wih a sraigh line, passing hrough he origin in C and supporing n(l) poinsofhepoinsea A, where n(l) is he number of realizaions of he raio l = y x : x, y A. As we ofen refer

4 ON NEW SUM-PRODUCT-TYPE ESTIMATES 975 o lines hroughou he paper, we use he symbol l o denoe individual members of he raio se. Even hough he proofs of Theorems 1.1 and 1. are essenially differen, he popular subse P A A is defined in he same way for boh heorems. Ye P denoes differen poin ses apropos of he raio and produc se cases, which figure wihin each heorem and are described nex. The same holds for he noaions L, N peraining o he poin se P. In paricular, he noaion L refers o he se of he corresponding popular raios, or lines hrough he origin. Raio se case. In order o esablish he esimaes involving he raio se, he noaion L will sand for he se of lines hrough he origin in C, supporing a leas 1 A A : A 1 poins of A A each. The subse P of A A suppored on hese popular lines is hen such ha P 1 A. (Indeed, he lines ouside L suppor a mos 1 A A : A 1 A : A = 1 A poins.) The noaion N will be used for he maximum number of poins per line in L. Trivially, N A, and one has A / L A : A. Produc se case. In order o esablish he esimaes involving he produc se, he same noaions P, L, N will be used for slighly differenly defined, muliplicaive, energy-based quaniies. The muliplicaive energy E (A) ofa is defined as follows: E (A) = {(a 1,...,a 4 ) A A : a 1 /a = a 3 /a 4 }. Since he equaion defining E (A) can be rearranged as a 1 a 4 = a a 3, by he Cauchy Schwarz inequaliy one has (.1) E (A) A 4 A A. Geomerically, E (A) is he number of ordered pairs of poins of A A C, suppored on sraigh lines hrough he origin, whose slopes l are members of he raio se A : A. A line is idenified by is slope l (which is well defined, since 0 A) and suppors some number n(l) poinsofa A. By he pigeonhole principle, here exiss some N [1,..., A ] such ha if L denoes he se of all lines wih N <n(l) N, hen (.) L N E (A) log A A 4 A A log A. (Indeed, i suffices o consider only dyadic values of N =1,,..., j,..., wih j = O(log A ), since rivially n(l) A.) Now, in he produc se case, le P be a popular muliplicaive energy subse of A A, conaining all poins of A A, suppored on he lines in he above defined se L, saisfying (.). The quaniy N gives he maximum, as well as he approximae number of poins of P per line l L, hais, P L N. (This approximae equaliy means ha P L N and L N P.) 3. Proof of Theorem 1.1. Wihou loss of generaliy, as we are no pursuing opimal consans in he esimaes, we may assume ha he se A C\{0} is locaed in a reasonably small angular secor, of angular half-widh an( arg z) <ɛaround he real axis, wih he verex a 0, so ha in paricular 0 A+A. The consan ɛ>0 does no go o zero: i needs only o be small enough for he geomeric argumen in he end of he proof of he forhcoming claim o be valid. One can amply se ɛ =

5 976 SERGEI V. KONYAGIN AND MISHA RUDNEV Theorem 1.1 will follow from he following claim. Claim. Le l 1,l be wo disinc members of he raio se (A : A) C = R,wih some realizaions l 1 = y1 x 1 and l = y x for x 1,y 1,x,y A. Consider l 1,l as poins in R. Then he poin z = y1+y x 1+x lies in C = R in some open se M (l1,l ), conaining he open sraigh line inerval (l 1,l )={l 1 +(1 )l, (0, 1)} and symmeric wih respec o his line inerval. Furhermore, consider he raio se as a verex se of a ree T in R, and le he sum of he Euclidean lenghs of he edges of T be minimum; i.e., le T be a minimum spanning ree on he verex se A : A. Then,if(l 1,l ) runs over he edges of T (wefurherwriesimply(l 1,l ) T ), he ses M (l1,l ) are pairwise disjoin. The above claim represens a bona fide generalizaion of he consrucion of Solymosi [16] for he posiive reals. Here is how he claim applies o he posiive real case. The se A : A lies on he posiive real axis. The edges of is minimum spanning ree are consecuive open line inervals beween he verices, and he ses M (l1,l ) are hese inervals hemselves. In he forhcoming proof of he claim we will describe he open ses M (l1,l ) precisely. Through he res of his secion we assume he claim and show how i resuls in Theorem 1.1, by essenially repeaing he argumen in [16]. Indeed, suppose ha here are respecively n(l 1 )andn(l )disincrepresenaions of some wo fixed raios l 1,l A : A; hais,l i = yj i i,x ji x j i i A, y ji i A, i for i = 1, and j i = 1,...,n(l i ). From basic linear algebra, he vecor sums (x j1 1 + xj,yj1 1 + yj ) C aain n(l 1 )n(l ) disinc values for disinc (j 1,j ). Assuming he claim, on he oher hand, ells one ha for all (j 1,j )heraio yj 1 1 +y j x j 1 1 +x j C = R lies in he se M (l1,l ). Now he fac ha he open ses M (l1,l ) are pairwise disjoin implies ha he map (x 1,y 1 ) (x,y ) (x 1 + x,y 1 + y ) (3.1) for x 1,y 1,x,y A : (l 1,l ) T, wih y 1 = l 1, y = l, x 1 x is an injecion. Indeed, assuming he conrary suggess ha here is a pair of disinc edges, (l 1,l )and(l 1,l )ofhereet, such ha (x 1 + x,y 1 + y )=(x 1 + x,y 1 + y ), where l 1 = y1 x 1, l = y x, l 1 = y 1 x, l = y y 1 x. Then, clearly, 1+y x 1+x = y 1 +y x,which 1 +x conradics he claim ha y1+y x 1+x and y 1 +y x lie, respecively, in he open ses M (l1,l 1 +x ) and M (l 1,l ), which are pairwise disjoin. The injeciviy of he map (3.1) accouns for he following inequaliy: (3.) A + A n(l 1 )n(l ) 1 (n(l 1 )+n(l )) min(n(l 1 ),n(l )). (l 1,l ) T (l 1,l ) T The inequaliy (3.) clearly remains rue if one resrics he verex se of T o any subse of A : A wih more han one elemen, in which case T will be a minimum spanning ree buil on hese verices. I is a his poin when one has o disinguish beween he raio and produc se cases by considering as verices of T only he raios from he popular se L, defined relaive o he raio or produc se case in secion. Given he se of verices L, let be a minimum spanning ree buil on he verex se L in R.ThusT has L verices and L 1edges.

6 ON NEW SUM-PRODUCT-TYPE ESTIMATES 977 In he raio se case, one has (3.3) A + A A 4 A : A (l 1,l ) T In he produc se case, where N (3.) and (.), ha (3.4) A + A ( L 1) N A A:A n(l) A l L, and hus, from (3.), (n(l 1 )+n(l )) A 4 A : A n(l) l L A 4 A : A. n(l) N A l L, he claim implies, by 4 E (A) log A A 4 A A log A, hus proving he second inequaliy in (1.3). (In view of (.), one can assume ha L > 1, for oherwise A A is large enough o ensure (1.3) immediaely.) This complees he proof of Theorem 1.1, condiional on he claim Proof of he claim. Suppose ha x 1,x,y 1,y A, Then, wih u = x /x 1,wehave y 1 x 1 = l 1, y x = l. y 1 + y (3.5) = y 1 + y x 1 + x x 1 (1 + u) = l 1 1+u + l u 1+u = l u 1 +(l l 1 ) 1+u. Since we have assumed ha an argx 1, an argx <ɛ,clearly u lies in he open u angular wedge W ɛ = {z :an arg z <ɛ}, and herefore 1+u lies in he image of W ɛ, furher denoed as M ɛ, under he Möbius map z = z 1+z. A sraighforward calculaion shows ha M ɛ is an open meniscus around he real line inerval (0, 1). The meniscus is formed by he inersecion of wo open discs cenred respecively a z ± =( 1, ± ι ɛ ), wih equal radii z ±. I is clearly symmeric around is major axis, ha is, he real line inerval (0, 1). The boundary of each disc inersecs he major axis a he angle whose angen equals ɛ, he half-widh of W ɛ. Clearly, M ɛ is amply conained in he open rhombus, whose major diagonal connecs he zero wih 1, and he minor diagonal has lengh ɛ. The meniscus M ɛ defines an open se M (l1,l ) menionedinheclaimasacomposiion of a dilaion and a ranslaion of M ɛ : by (3.5), y 1 + y M (l1,l x 1 + x ) = {l 1 +(l l 1 )M ɛ }. Thus, he se M (l1,l ) is conained in he open rhombus wih main diagonal denoed as e =(l 1,l ) and minor diagonal lengh ɛ l l 1. This rhombus will be furher denoed as R e = R (l1,l ). Through he res of his secion, le L be any nonempy subse wih more han one elemen of he raio se A : A. Le T be a minimum spanning ree buil on he verex se L. Thais, T has he minimum ne Euclidean lengh of he edges over all he rees wih he verex se L. The ree T has L 1 edges, which are open sraigh line segmens connecing some pairs of disinc verices in he se L. There are no loops in T, and for any pair of disinc verices l 1,l L here is a unique pah connecing hem. Through he res of his secion, we will use he uppercase Lain leers A,B,C,D,...for he verices of T, in conras o he res of he paper, where A,B,... are ses. Firs, noe he well-known fac ha T may no conain inersecing edges. Indeed, suppose ha (AB) and(cd) areedgesoft and (AB) (CD). InhereeT hereisauniquepahfromb o C and a unique pah from B o D. SinceT has no

7 978 SERGEI V. KONYAGIN AND MISHA RUDNEV loops, one of hese wo pahs, wihou loss of generaliy he one from B o D, mus conain he edge (CD)(forif(CD) is no conained in eiher of he wo pahs, here is a pah from C o D oher han (CD), via B). Then he pah from eiher A or B o D conains boh edges (AB) and(cd). Wihou loss of generaliy, le i be he pah connecing A and D. Thus, if (AB) (CD), hese edges can be deleed and replaced by he edges (AC) and(bd), wihou violaing conneciviy or creaing loops. On he oher hand, [AC] and[bd] are a pair of opposie sides of he convex quadrilaeral ACBD, while [AB] and[cd] are is diagonals. Bu he sum of he lenghs of eiher pair of opposie sides of a convex quadrilaeral is smaller han he sum of he lenghs of he diagonals. This conradics he minimaliy of T. In a minimum spanning ree he angle beween adjacen edges is a leas π 3.Tosee his fac, suppose ha here are wo edges (AB) and(ac), wih he angle beween hem a A smaller han π 3. Then one of he wo remaining angles in he riangle ABC exceeds π 3, and he edge opposie o i in T can be deleed and replaced by he shorer edge (BC) wihou violaing conneciviy or creaing loops. This conradics he minimaliy of T. Therefore, he rhombi around adjacen edges canno inersec, because he angen of he half-angle of R (AB) a A or B is jus ɛ. The supposiion ha he rhombi around a pair of adjacen edges (AB) and(ac) inersec would conradic he fac ha he angle beween hem a A is smaller han π 3. Finally, suppose ha here is a pair of nonadjacen and noninersecing edges (AB) and(cd) such ha R (AB) R (CD). Le us show ha his also leads o a conradicion if ɛ is small enough. The key observaion is he following lemma. Lemma 3.1. The verices C, D canno lie in he open disk wih diameer (AB). Proof. Suppose ha, say, C lies inside he open disk wih he diameer (AB). Then he angle ACB is obuse. Hence, he edge (AB) can be deleed and replaced in he ree T by one of he shorer line segmens (AC) or(bc), wihou violaing conneciviy or creaing loops. More precisely, if he unique pah from A o C in T incorporaes (AB), hen (AB) should be replaced by (AC), and oherwise by (BC). This conradics he minimaliy of T. Le us use Lemma 3.1 ogeher wih he fac ha (AB) (CD)= for he proof of he following lemma. Lemma 3.. If R (AB) R (CD) and α is he angle beween (AB) and (CD), ɛ 1 ɛ. hen an α Proof. Firs we assume ha (CD) inersecs he rhombus R (AB). By Lemma 3.1, neiher C nor D belongs o he closure of R (AB). Hence, (CD) inersecs he boundary of he rhombus R (AB) a wo poins, say E and F. Nex, since [EF] (CD) does no inersec (AB), we conclude ha he angle α beween [EF] and(ab) saisfies he inequaliy an α<ɛasrequired. Similarly, we prove our asserion if (AB) inersecs R (CD). Now we consider he case where (CD) does no inersec R (AB) and (AB) does no inersec R (CD). Then he boundaries of he rhombi R (AB) and R (CD) have wo common poins, say E and F.Thesegmen[EF] does no inersec he edges (AB) and (CD). Therefore, he angle α 1 beween [EF]and(AB) and he angle α beween [EF] and(cd) saisfy he inequaliies an α 1 <ɛand an α <ɛ.leαbe he angle beween (AB) and(cd). Then we have α α 1 + α, and he asserion of he lemma follows.

8 ON NEW SUM-PRODUCT-TYPE ESTIMATES 979 Finally, o refue he assumpion R (AB) R (CD), assume, wihou loss of generaliy, ha AB =1, AB CD, A=0,andB = 1. Le us now use he conclusion ha (AB) and(cd) are close o being parallel, along wih Lemma 3.1 for a rough esimae as o where he verices C, D can be locaed. They may no lie inside he open disc wih he diameer (AB). Since R (AB) R (CD), CD AB, and an α ɛ 1 ɛ,whereαis he angle beween (AB) and(cd), neiher C nor D may possess an imaginary par whose absolue value is in excess of 4ɛ. If ɛ is small enough, he real par of he lefmos poins, where horizonal lines wih Iz =4ɛ inersec he circle wih he diameer AB =1,isO(ɛ ). Hence, since CD AB, conclude ha one of he endpoins of (CD), say C, mus lie inside he open square box {max( Rz, Iz ) < 4ɛ} around A, andd inside he same box ranslaed by 1, so ha is cener is now B. This, once again, implies conradicion wih he minimaliy of T. Indeed, now if ɛ is small enough, he edge (AB), whose lengh is 1, can be deleed in T and replaced by a shorer edge (AC) or(bd), wihou violaing conneciviy or creaing loops. As in he proof of Lemma 3.1, he replacemen will be (AC) if he unique pah from A o C in T incorporaes (AB), and (BD) oherwise. We have exhaused all he possibiliies for he muual alignmen of a pair of edges (AB) and(cd). Thus for wo disinc edges e 1,e T, he open rhombi R e1 and are disjoin. This complees he proof of he claim. R e 4. Proof of Theorem Lemmaa. The main ool o prove Theorem 1. is he Szemerédi Troer incidence heorem. For any se P of poins and any se L of sraigh lines in a plane le I(P, L) ={(p, l) P L: p l} be he se of incidences. Theorem 4.1 (Szemerédi and Troer [17]). The maximum number of incidences in R is bounded as follows: (4.1) I(P, L) ( P L ) 3 + P + L. As a resul, if P (or L ) denoes he ses of poins (or lines) inciden o a leas 1 lines(orpoins)ofl (or P), hen (4.) P L 3 L P 3 + L, + P. Leusnoehahelinearin P, L erms in he esimaes (4.1), (4.) are essenially rivial and usually of no ineres in he sense of being dominaed by he nonlinear ones, whenever hese esimaes are being used. This is also he case in his paper. The Szemerédi Troer heorem is also rue in full generaliy in he plane over C. This was proved by Tóh [19]. A more modern proof came ou in a recen paper of Zahl [0]. In a paricular case, where he poin se is a Caresian produc, Solymosi [15, Lemma 1], observed ha he proof of he C version of he Szemerédi Troer heorem is considerably more sraighforward han dealing wih arbirary poin ses in C. Alhough he geomeric par of he forhcoming proof closely follows he

9 980 SERGEI V. KONYAGIN AND MISHA RUDNEV consrucion in [15], he poin ses o which we apply he heorem are no necessarily Caresian producs, so sricly speaking, we are using here he general version of he Szemerédi Troer heorem in C of Tóh and Zahl. The esimaes (4.) will be furher used in he C seing wihou addiional commens. One can easily develop a weighed version of he esimaes of he Szemerédi Troer heorem, quoed nex (see Iosevich e al. [5]). Suppose ha each line l L has been assigned a weigh m(l) 1. The number of weighed incidences i m (P, L) is obained by summing over he se I(P, L), each pair (p, l) I(P, L) being couned m(l) imes. Suppose ha he oal weigh of all lines is W and he maximum weigh per line is μ>0. Theorem 4.. The maximum number of weighed incidences beween a poin se P and a se of lines L, wih he oal weigh W and maximum weigh per line μ, is bounded as follows: (4.3) i m (P, L) μ 1 3 ( P W ) 3 + μ P + W. The second main ingredien for proving Theorem 1. comes from a purely addiivecombinaorial observaion by Schoen and Shkredov [11, Lemma 3.1], which has recenly allowed for several incremenal improvemens owards a number of open quesions in field combinaorics in [11], [1], [14]. This observaion is he conen of he following Lemma 4.3, he quoing of which requires some noaion also o be used in wha follows. Through he res of his secion A, B denoe any ses in an Abelian group (G, +). In he conex of he field C, Lemma 4.3 will apply o he addiion operaion, so he noaion E will sand for he addiive energy, raher ha he muliplicaive energy E, which has been used in he proof of he sum-produc esimae in Theorem 1.1. For any d A A, se (4.4) A d = {a A : a + d A}. The quaniy E(A, B) = {(a 1,a,b 1,b ) A A B B : a 1 a = b 1 b } is referred o as he addiive energy of A, B. By he Cauchy Schwarz inequaliy, rearranging he erms in he above definiion of E(A, B), one has (4.5) E(A, B) A ± B A B. Indeed, if d or x is, respecively, an elemen of A B or A + B, andn(d) orn(x) is he number of is realizaions as a difference or sum of a pair of elemens from A B, i.e., n(d) = {(a, b) A B : d = a b}, he esimae (4.5) follows from he fac ha (4.6) E(A, B) = n (d) = d A B x A+B n (x). The quaniy E(A, A) =E(A) is referred o as he (addiive) energy of A. Noeha, according o (4.4), n(d) = A d for d A A.

10 ON NEW SUM-PRODUCT-TYPE ESTIMATES 981 We will also need he cubic energy of A, defined as follows: (4.7) E 3 (A) = {(a 1,...,a 6 ) A A : a 1 a = a 3 a 4 = a 5 a 6 }. This definiion implies (see [1, Lemma ]) ha (4.8) E 3 (A) = E(A, A d ). d A A To see his, le us wrie d A A all quadruples saisfying (4.9) a 1 a 3 = a a 4 wih a 1,a A, a 3,a 4 A d. The lis will conain d A A E(A, A d) quadruples. Any quadruple is repeaed as many imes as here are many differen values of d wih a 3,a 4 A d. This number is he number of pairs (a 5,a 6 ) wih a 5,a 6 A and (4.10) a 5 a 3 = a 6 a 4. Bu he number of collecions (a 1,...,a 6 ) of elemens from A saisfying boh (4.9) and (4.10) is jus E 3 (A). The following saemen is par of Corollary 3 in [1]. Since he formulaion we use is slighly differen from he original one and in an effor o make his paper self-conained, we have chosen o include is proof as well. Lemma 4.3. Le A be a finie nonempy addiive se. For any D A A, one has (4.11) ( A d D A d 3 A d A A d E 3 (A) d D Proof. To verify (4.11) observe ha, by he inequaliy (4.5) applied o he ses A, A d for a fixed d, wehave A Ad E(A, A d ) A A d. Muliplying boh sides by A d and summing over d D, hen applying once again he Cauchy Schwarz inequaliy o he lef-hand side, yields A d A A d E(A, A d ) A A d 3. d D d D d D Squaring boh sides and using (4.8) complees he proof of Lemma 4.3. Corollary 4.4. Le A be a finie nonempy addiive se. For any D A A, one has ( ) (4.1) E(A, A A)E 3 (A) A A d 3. d D Proof. The proof is based on an observaion ha [1] credis o Kaz and Koeser (see [7]) ha he lef-hand side of (4.11) provides a lower bound for E(A, A A). Indeed, each d A A has A d represenaions d = u v wih u A, v A d.the ).

11 98 SERGEI V. KONYAGIN AND MISHA RUDNEV same d also has a leas A A d represenaions d = u v wih u, v A A. Indeed, given d, for any v A d and a A one can find u A so ha d =(u a) (v a), wih A A d disinc values for he second bracke. Hence, if n(d) ishenumber of represenaions of d as an elemen of A A and n (d) is is represenaions as an elemen of (A A) (A A), hen E(A, A A) = n(d)n (d) A d A A d. d d This, ogeher wih (4.11) complees he proof of Corollary 4.4. Remark 4.5. In he forhcoming main body of he proof of Theorem 1. we will use he Szemerédi Troer heorem o yield upper bounds for he wo energy erms in he lef-hand side of he esimae (4.1) for he addiive poin se P C, defined in secion for he raio and produc se cases. From now on, le he above D A A be a popular subse of he difference se A A, defined as follows: { (4.13) D = d A A : A d 1 A }. A A Then, since d (D ) A d 1 c A, ( ) 1 A A d 3 A d 1 A A 4 d D d D ( A A A ) 1 A. Subsiuing his ino he saemen of Corollary 4.4, le us formulae he resul as our final corollary, which summarizes he above-menioned arihmeic componen of he argumen. Corollary 4.6. Le A be a finie nonempy addiive se. Then (4.14) E 3 (A)E(A, A A) A 8 A A. We conclude his preliminary secion wih a remark discussing some recen applicaions of Lemma 4.3 and is corollaries. The conen of he remark is no used direcly in he main body of he proof of Theorem 1.. Remark 4.7. The esimae (4.14) enabled Schoen and Shkredov [1] o achieve progress on he sum se of a convex se problem, see [3]. They proved ha if A = f([1,...,n]), where f is a sricly convex real-valued funcion, hen A A A 8 5 log 5 A, having improved he previously known exponen 3. The conjecured exponen in he sum se of a convex se problem is, modulo a facor of log A. Li [8] see also his recen work wih Roche-Newon [9] poined ou ha he approach of [1] can be adaped o he sum-produc problem, using a varian of he well-known sum-produc consrucion by Elekes []. This improves he exponen 5 4 obained by Elekes wihin his consrucion o 14 11, modulo a facor of log A. The same exponen 14 11, modulo a facor of log A, had been coincidenally obained in Solymosi s work [16], as saed in (1.) above. Also recenly Jones and Roche-Newon [6] applied he esimae (4.14) o improve he bes known lower bound on he size of A(A +1) in he real seing. (The laer paper also conains a new lower bound on A(A +1) in a finie field seing, obained via a differen echnique.) 4.. The main body of he proof of Theorem 1.. Recall he definiion of he poin se P, as well as he quaniies L, N in he end of secion, relaive o

12 ON NEW SUM-PRODUCT-TYPE ESTIMATES 983 eiher he raio or produc se case. In eiher case, le us consider he vecor sum se of he se P C wih some poin se Q such ha Q P. (In wha follows we will se Q = P or P P.) Recall ha he se P conains all poins of A A suppored on a popular se of lines hrough he origin L. To obain he vecor sums, one ranslaes he lines from L o each poin of Q, geing hereby some se L of lines wih L L Q. In boh he raio and produc se cases, i can be assumed ha (4.15) L 1 N. The esimae (4.15) is clear in he raio se case, where N A L. As o he produc se case P L N, we will need he following lemma, which will be used once more in he end of he proof of Theorem 1.. Lemma 4.8. There exis L, N saisfying (.) and such ha (4.16) N A A A A A 3. A varian of Lemma 4.8 can be found in he recen papers [9], [13] and represens a sligh generalizaion of he well-known approach o he sum-produc problem due o Elekes []. The proof of Lemma 4.8 is given in he final secion of he paper. The bound (4.16) for N and he fac ha LN is bounded from below by (.) would yield under he assumpion L N ha Therefore, A A 6 A A 3 A 9 LN A A 6 A A 4 A 13 log A, A 4 A A log A. which is beer han (1.4). Thus we assume he esimae (4.15) henceforh. We reurn o analyzing he se of lines L. The Szemerédi Troer heorem enables one o esimae L from below. We have he following esimae for he number of incidences (4.17) L Q I(Q, L) L 3 Q 3 + L + Q. Since i can be assumed ha L is bigger han some absolue consan (as he arge esimaes (1.4) are up o absolue consans), he erm Q in (4.17) canno dominae he esimae. Nor can he erm L, for oherwise L > Q. This, since by consrucion of L one has L L Q, would imply L > Q, bu in our seup Q P L. Thus i follows from (4.17) ha (4.18) L L 3 Q 1. Le us call he number of poins of Q on a paricular line l L,heweighm(l) ofl. The oal weigh W of all lines in he collecion L is by consrucion equal o L Q.

13 984 SERGEI V. KONYAGIN AND MISHA RUDNEV Le us now sudy he se P + Q. The vecor sums in P + Q are obained by he parallelogram rule; hence we observe ha P + Q is suppored on he union of he lines from L, assubsesofc : (4.19) P + Q l. l L Our goal now is o obain upper bounds, in erms of 1, on he number of elemens of P +Q, whose number of realizaions as a sum p + q : p P, q Q is a leas. The same line l Lcan conribue o he same vecor sum x = p + q P +Q, q l, a mos min(n,m(l)) imes. In view of his, we can lower he weighs of lines which are oo heavy : whenever m(l) N, le us redefine i as N. Afer his has been done, W denoing he oal weigh of he lines in L, one has (4.0) W L Q. Also, we define (4.1) m = Q L. The Szemerédi Troer heorem, namely (4.), ells one ha he weigh disribuion over L obeys he inverse cube law; i.e., for N one has (4.) L = {l L: m(l) } Q 3 + Q Q 3, as since N L N P Q, he rivial erm Q ges dominaed by he firs erm. I also follows from (4.), via he sandard dyadic summaion in, ha he oal weigh W (L ) suppored on he lines from L is bounded by (4.3) W (L ) Q. (To see his one pariions L ino dyadic subses of lines whose weighs τ N are j τ< j+1 for j 0. I follows from (4.) ha W (L ) Q j 0 j+1.) Suppose, in view of (4.19), ha some x P + Q is inciden o k 1 lines l 1,...,l k L.Wehenhaveaninequaliy (4.4) n(x) m(x), where (4.5) n(x) = {(p, q) P Q : x = p + q}, m(x) = k m(l k ). Observe now ha if n(x) >N,henx P + Q mus be inciden o more han one line from L. Indeed, each line l Lmay conribue a mos N o he quaniy n(x). Hence, le P(L) denoe he se of all pairwise inersecions of lines from L. We can herefore bound he maximum number of poins in P + Q, whosenumberof realizaions n(x) is a leas >N, in erms of, by way of bounding he number of i=1

14 ON NEW SUM-PRODUCT-TYPE ESTIMATES 985 x P(L), wih m(x). The laer bound will follow from Theorem 4. ogeher wih he inverse cube weigh disribuion bounds (4.), (4.3) over he se of lines L. Namely, we have he following lemma, which also has is prooype in [5, Lemma 6]. Lemma 4.9. Suppose ha Q P. Then for some absolue C and : CN P, (4.6) {x P + Q : n(x) } L 3 Q 5 3, Proof. Observe ha for any poin se P he number of weighed incidences i m (P, L) ofl wih P can be bounded from above using dyadic decomposiion of L by weigh in excess of m, as follows: (4.7) i m (P, L) log N/ m j=0 i m (P, L j m). Above, he noaion L m, corresponding o j = 0, sands for he subse of L conaining all hose lines whose weigh does no exceed m, and L j m = {l L: j 1 m <m(l) j m}, j 1. To esimae each individual erm i m (P, L j m) in he sum (4.7), one can use he esimae (4.3) of Theorem 4.. The quaniy j m hen replaces he maximum weigh μ in (4.3). The oal weigh W in (4.3) will be replaced by he oal weigh W j m of he line se L j m. In view of (4.3), he quaniy W j m is bounded as follows: (4.8) W j m Q j m = Q L j. Thus (4.9) i m (P, L j m) ( j m) 1 3 ( P Q L j ) 3 +j m P + W j m. Using (4.8), i follows ha in he summaion (4.7), he erm j = 0 dominaes he ne conribuion of he firs and he hird erms in he esimae (4.9) for j> 0. Conversely, he dominan value of he second erm in (4.9) corresponds o he maximum value N of he lines weigh. Thus (4.30) i m (P, L) m 1 3 ( P W ) 3 + N P + W P 3 L 1 Q N P + L Q, using (4.0). Recall ha in view of (4.4), for >N we have he inclusion (4.31) {x P + Q : n(x) } (P {p P(L) : m(p) }). Hence, we apply he incidence bound (4.30) o he poin se P, ogeher wih he lower bound (4.3) P i m (P, L).

15 986 SERGEI V. KONYAGIN AND MISHA RUDNEV I follows ha for CN, where he consan C is deermined by he consans hidden in he symbol in he esimae (4.30), he second erm on he righ-hand side of he esimae (4.30), applied o he se P, canno possibly dominae he esimae. Thus, for CN N, one has (4.33) P L 3 Q 5 I follows ha 3 + L Q. (4.34) P L 3 Q 5 3 for CN 4 L Q 3. For larger, one has o be slighly more careful wih he erm L Q in (4.33), which, in fac, can be refined for (4.35) L m = Q L 4 L Q 3. Noe ha he lines in L come in L possible direcions, and herefore no more han L lines can be inciden o a single poin in P(L). Hence, lines from a dyadic se L j m canno conribue more han a small proporion o he oal number of weighed incidences suppored on he ses P if is much greaer han L ( j m). More precisely, suppose ha = L ( i m), i 1. I follows ha for such he esimae (4.7) can be resaed for he se P as follows: (4.36) log 1 N/ m P i m (P, L j m). j=i Indeed, he oal conribuion of he dyadic ses L j m o he quaniy m(x) for x P and j<iis a mos i 1 L m =. We now repea he argumen esimaing he righ-hand side, which has led from (4.7) o (4.30), having in mind ha i is only he las erm W in he firs line of (4.30) ha needs o be changed. Namely, W should ge replaced by he oal weigh of he lines, conribuing o he righ-hand side of (4.36). These are he lines whose individual weigh is a leas L. Le W denoe he oal weigh suppored on L hese lines. By (4.3) we can esimae W Q L L. Thus for L m he esimae (4.33) can be improved as follows: (4.37) P L 3 Q Q L 3. Since Q P L, he firs erm in (4.37) dominaes he esimae, and in view of (4.35), one has (4.38) P L 3 Q 5 3 for 4 L Q 3. The esimaes (4.34) and (4.38) and he inclusion (4.31) now complee he proof of Lemma 4.9.

16 ON NEW SUM-PRODUCT-TYPE ESTIMATES 987 All he key ingrediens for finishing he proof of Theorem 1. have been developed. We now use Lemma 4.9 o yield an upper bound for he lef-hand side in he esimae (4.14) of Corollary 4.6, applied o he addiive se P. Using Lemma 4.9 wih Q = P, we can bound he quaniy E 3 (P ) as follows: (4.39) E 3 (P )= x P P log A n 3 (x) N P + L 3 5 P 1. Above, he firs erm deals wih he se of all x P P, whose number of realizaions n(x) is less han he applicabiliy hreshold = CN of Lemma 4.9, wih some absolue consan C. Inoherwords, n 3 (x) N n(x) N P. x P P : n(x)<cn x P P The second erm in (4.39) resuls from applying Lemma 4.9 o he par of he cubic energy suppored on {x P P : CN n(x) P }, using dyadic summaion. Namely, for j 0leX j = {x : j CN n(x) < j+1 CN P }. Then n 3 (x) X j ( j+1 CN) 3, x P P : CN n(x) P j 0 and since X j is nonempy for j = O(log A ) only, he bound (4.6) for X j,where one ses = j CN, resuls in he second erm in (4.39). In boh he raio and produc se cases, by (4.15), N 4L 4 P L 3. Thus he second erm dominaes he esimae (4.39); ha is, (4.40) E 3 (P ) L 3 P 5 log A. Subsiuing he esimae (4.40) ino (4.14) yields P 11 (4.41) E(P, P P ) L 3 P P log A. Now one can also use Lemma 4.9 wih Q = P P o esimae he quaniy E(P, P P ) from above. I follows from (4.6) ha for any CN one has (4.4) E(P, P P ) P P P + L 3 P P 5. Above, he firs erm gives a rivial bound for he conribuion o E(P, P P )ofall hose x P + P P which have fewer han realizaions n(x). The second erm uses (4.6) and bounds he conribuion o E(P, P P ) of he erms wih or more realizaions: his conribuion is bounded by he dyadic sum {x P + P P : n(x) j } ( j+1 ) L 3 P P 5. j=0 Now we can choose = C P P 3 4 L 3 4 P, j=0

17 988 SERGEI V. KONYAGIN AND MISHA RUDNEV where he consan C is large enough o ensure ha CN, he applicabiliy hreshold of Lemma 4.9. Such a C exiss, since P P P L N. The above choice of in (4.4) yields (4.43) E(P, P P ) P P P 7 4 L 3 4. Combining his wih (4.41) resuls in he following inequaliy: (4.44) P P 11 9 P 5 4 L 4 log A. I remains o eliminae L from he laer esimae, relaive o he raio or he produc se case. To obain he firs esimae of (1.4) as o he raio se case, i suffices o noe ha P 1 A, L A : A, as well as P P A A. In he produc se case, where P L N, he esimae (4.44) becomes (4.45) A A 11 ( L N ) N log A The quaniy LN is bounded from below by (.), and Lemma 4.8 provides a nonrivial upper bound (4.16) for N. Subsiuing hese bounds ino (4.45) yields he second esimae of (1.4) and complees he proof of Theorem Proof of Lemma 4.8. A varian of Lemma 4.8 can be found in he recen papers [9], [13] and represens a sligh generalizaion of he well-known approach o he sum-produc problem due o Elekes []. For compleeness sake, we furher presen a simple proof. The noaion in he forhcoming argumen is somewha independen from he res of he paper. Consider a se A, no conaining zero, and a se of lines L = {y = d+x a },where d is an elemen of he difference se A A and a A. Clearly here are A A A lines. Therefore, he number of poins in a se P, where more han lines from L inersec, is, by (4.), bounded as follows: (4.46) P A A A + Suppose now ha 3 A A A. L = {l A : A, n(l) >}. For each l L one has l = a i a i, where he index i runs over n(l) disinc values. Given l L, for every a A one has l = (a i a)+a a i for i =1,...,n(l); i.e., he poin in he plane wih coordinaes (a, l) isincidenoaleasn(l) lines from L, hese lines being idenified by he pairs (d i = a i a, a i), wih i =1,...,n(l). Hence A L P, and i follows from (4.46) ha (4.47) L A A A 3 + A A. Le us use (4.47) o esimae he conribuion of he se L A : A o he muliplicaive energy E (A). For j =0, 1,..., wih he upper bound j+1 A, he se of raios {l A : A, j <n(l) j+1 A } conribues o E (A) amos 4 L j ( j ) A A A j + A A ( j ).

18 ON NEW SUM-PRODUCT-TYPE ESTIMATES 989 Summing he righ-hand side over j yields a bound for he conribuion of he se L o he muliplicaive energy E (A), as follows: l A:A, <n(l) A n (l) A A A + A A A A A A. Comparing his wih he lower bound (.1) for E (A) shows ha for some C one can se A A A A (4.48) = C A 3 and have he following inequaliy: l A:A, n(l) n (l) 1 A 4 A A. Thus, here exiss a dyadic subse of {l A : A, n(l) }, namely he se L = {l A : A, N <n(l) N}, for some N, such ha his se L conribues 1 A o he muliplicaive energy E (A) aleasheamoun 4 log A A A. Since saisfies (4.48), his proves Lemma 4.8. Remark The argumen in he above proof of Lemma 4.8 is symmeric wih respec o he wo field operaions in C: by defining he se of lines as L = {y = lx a}, where (l, a) (A : A) A, one can ge a similar upper bound on he maximum number of realizaions of popular differences (or sums, by a rivial modificaion) conribuing o he addiive energy E(A), via he raio or produc se. 5. Acknowledgmen. The second auhor hanks Oliver Roche-Newon for helpful discussions and remarks. REFERENCES [1] T. Bloom and T. Jones, A Sum-Produc Theorem in Funcion Fields, preprin, arxiv: , 01. [] G. Elekes, On he number of sums and producs, Aca Arihmeics, 81 (1997), pp [3] G. Elekes, M. Nahanson, and I. Z. Ruzsa, Convexiy and sumses, J. Number Theory, 83 (000), pp [4] P. Erdős and E. Szemerédi, On sums and producs of inegers, in Sudies in Pure Mahemaics, Birkhäuser, Basel, 1983, pp [5] A. Iosevich, S. Konyagin, M. Rudnev, and V. Ten, Combinaorial complexiy of convex sequences, Discree Compu. Geom., 35 (006), pp [6] T. Jones and O. Roche-Newon, Improved bounds on he se A(A +1), J.Combin.Theory Ser. A, o appear. [7] N. H. Kaz and P. Koeser, On addiive doubling and energy, SIAM J. Discree Mah., 4 (010), pp [8] L. Li, On a Theorem of Schoen and Shkredov on Sumses of Convex Ses, preprin, arxiv: , 011. [9] L. Li and O. Roche-Newon, Convexiy and a Sum-Produc Type Esimae, preprin, arxiv: , 011. [10] M. Rudnev, An improved sum-produc inequaliy in fields of prime order, In. Mah. Res. Noices, 16 (01), pp [11] T. Schoen and I. Shkredov, Addiive properies of muliplicaive subgroups of F p,quar.j. Mah., 63 (01), pp [1] T. Schoen and I. Shkredov, On sumses of convex ses, Comb. Probab. Compu., 0 (011), pp

19 990 SERGEI V. KONYAGIN AND MISHA RUDNEV [13] T. Schoen and I. Shkredov, Higher Momens of Convoluions, preprin, arxiv: , 011. [14] I. Shkredov and I. Vyugin, On addiive shifs of muliplicaive subgroups, Sb. Mah., 03 (01), pp [15] J. Solymosi, On he number of sums and producs, Bull. London Mah. Soc., 37 (005), pp [16] J. Solymosi, Bounding muliplicaive energy by he sumse, Adv. Mah., (009), pp [17] E. Szemerédi and W. T. Troer, Jr., Exremal problems in discree geomery, Combinaorica, 3 (1983), pp [18] T. Tao, The sum-produc phenomenon in arbirary rings, Conrib. Discree Mah., 4 (009), pp [19] C. D. Tóh, The Szemerédi-Troer Theorem in he Complex Plane, preprin, arxiv:030583, 003. [0] J. Zahl, ASzemerédi-Troer Type Theorem in R 4, preprin, arxiv: , 01.