#A27 INTEGERS 12 (2012) SUM-PRODUCTS ESTIMATES WITH SEVERAL SETS AND APPLICATIONS

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1 #A27 INTEGERS 12 (2012) SUM-PRODUCTS ESTIMATES WITH SEVERAL SETS AND APPLICATIONS Antal Balog Alfréd Rényi Institut of Mathmatics, Hungarian Acadmy of Scincs, Budast, Hungary Kvin A Broughan Dartmnt of Mathmatics, Univrsity of Waikato, Hamilton, Nw Zaland kab@waikatoacnz Igor E Sharlinski Dartmnt of Comuting, Macquari Univrsity, Sydny, Australia igorsharlinski@mqduau Rcivd: 10/7/11, Acctd: 3/18/12, Publishd: 3/23/12 Abstract W obtain svral vrsions of th sum-roduct thorm with k-fold sums and roduct sts W also giv svral alications of ths stimats 1 Introduction Lt F dnot th finit fild of lmnts For a st A F and a rational function F (X 1,, X m ) F (X 1,, X m ), which has no ols in A, w dfin th st F (A,, A) = {F (a 1,, a m ) : a 1,, a m A} In articular, for an intgr k, ka and A k dnot k-fold sums and roduct sts, rsctivly Th most intrsting and wll studid cas in th classical sum-roduct roblm whr th goal is to show that at last on of sts A 2 = A A and 2A = A + A is of siz substantially largr than A This dirction, initiatd by th ionring work of Bourgain, Katz & Tao [6], has bn dvlod in a various dirctions and has had svral imortant alications, s [2, 4, 5, 13, 14, 18, 23, 24, 25] and rfrncs thrin Hr, motivatd by som nw alications, w considr th cas of a k-fold sum and roduct sts A k and ka W not that svral rsults of ths tys for sts of intgr and ral numbrs hav bn givn by various authors, s [10] and rfrncs

2 INTEGERS: 12 (2012) 2 thrin For finit filds, thr ar also svral rsults for multil sum and roduct sts, s [10, 15, 16, 17], howvr this dirction has not yt bn systmatically studid Hr w rsnt svral gnral rsults of this ty In articular, w us th mthod of Garav [13], which in turn has its roots in th work of Elks [11] on th sum-roduct roblm ovr th rals, to show that for any intgr k 2 thr is a constant C > 0 such that } A k ka C min { A, A 2k k 1, which with k = 2 rcovrs [13, Thorm 1] W also giv svral nw alications For xaml, w imrov on of th stimats of [1] on th numbr of solutions of xonntial congruncs x x a (mod ), 1 x 1 (1) W us th following notations Throughout th ar, th imlid constants in th symbols O,, and may dnd on th intgr aramtrs k and ν and normally ar for through rims Rcall that th notations U V and V U ar quivalnt to U = O(V ) By (u) w man as usual x(2πiu/) If A is a finit st, A rrsnts th numbr of lmnts of A 2 Estimats from Arithmtic Combinatorics 21 Sum-Product Estimats W start with th following standard rsult on doubl xonntial sums, s [4, Bound (14)] Lmma 1 Lt b rim and A, B F Thn max (n,)=1 (nxy) A B x A y B W now rsnt th following siml modification of th rsult of Garav [13] Thorm 2 For arbitrary sts A, B, C F, with 0 B, w hav A B A + C 3 { } 8 min A, A 2 B C Proof As in [13], w considr th solutions of th quation whr S = A B and T = A + C s 1 + c = t, b B, c C, s S, t T, (2) b

3 INTEGERS: 12 (2012) 3 For any trilt (a, b, c) A B C thr is a uniqu solution, namly s = ab, t = a + c So (2) has at last A B C solutions On th othr hand, as in [13], using th bound for bilinar xonntial sums givn by Lmma 1 to stimat th total numbr of solutions to (2), on drivs A B C B C S T + 1 B S C T, (3) which imlis th rsult (in fact with th constant (3 5)/2 3/8) Corollary 3 For an arbitrary subst A F and intgr k 1, w hav A k ka min {c A, ck 1 A 2k }, whr c = 3/8 k 1 Proof W rov th dsird stimat by induction on k For k = 2, it is ssntially th rsults of [13] (and it also follows from Thorm 2 with B = C = A) Now, w assum that A k 1 (k 1)A min {c A, ck 2 A 2k 2 } k 2 Thn for k 3 w us Thorm 2 with B = A k 1 and C = (k 1)A, gtting { A k ka c min A, A 2 A k 1 } (k 1)A min {c A, c 2 A 3, ck 1 A 2k } k 1 Sinc for A < (/c) 1/2 th rsult is trivial (as c k 1 A 2k / k 1 A 2 ) and for A (/c) 1/2 w hav c A < c 2 A 3, th rsult now follows W also not that as in [26] on can us multilicativ charactr sums to stimat th numbr of solutions to (2) In articular w rcall a rsult of Karatsuba [21] (s also [22, Chatr VIII, Problm 9]), (which in turn follows from th Wil bound and th Höldr inquality) assrting that for a nontrivial multilicativ charactr χ modulo and arbitrary sts X, Y F w hav χ(x y) X 1 1/2ν Y 1/4ν + X 1 1/2ν Y 1/2 1/2ν, (4) x X y Y with any fixd intgr ν 1 With this, instad of th bound (3) w driv A B C B C S T + 1 ( ) B S T 1 1/2ν C 1/4ν + T 1 1/2ν C 1/2 1/2ν, which lads to anothr vrsion of Thorm 2 (that is strongr if C is small)

4 INTEGERS: 12 (2012) 4 Our scond aroach dnds on th following stimat of Bourgain & Garav [4, Thorm 12] Lmma 4 For arbitrary substs X, Y, Z F, as, (xyz) ( X Y Z )13/16 5/18+o(1) x X y Y z Z W ar now rady to rov th following stimat: Thorm 5 For arbitrary sts A, B, C, D F, w hav max{ A B C, A + D } min{ A, A 16/21 ( B C ) 1/7 D 8/21 40/189+o(1) } Proof W us a modification of th argumnt of Thorm 2 For th sts U = A B C and V = A + D, w considr th th numbr J of solutions (b, c, d, u, v) to th quation ub 1 c 1 + d = v, b B, c C, d D, u U, v V Clarly for a A, b B, c C, d D, th vctor (b, c, d, abc, a + d) is a solution Thus J A B C D (5) On th othr hand, w obviously hav J = 1 b B c C d D u U v V λ F ( λ(ub 1 c 1 + d v) ) Changing th ordr of summation, sarating th trm B C D U V / corrsonding to λ = 0, w obtain J = B C D U V + R, (6) whr R 1 ( λub 1 c 1) (λd) (λv) λ F u U b B c C By Lmma 4 w obtain 1 R ( B C U ) 13/16 13/18+o(1) (λd) (λv) (7) Alying Cauchy s inquality (and xtnding th summation ovr λ to F ) w driv 2 (λd) (λv) 2 2 (λd) (λv) λ F λ F d D v V λ=1 d D d D d D v V v V λ F v V

5 INTEGERS: 12 (2012) 5 Clarly and similarly (λd) λ F d D 2 = d 1,d 2 D λ F (λv) λ F v V 2 (λ(d 1 d 2 )) = D = V Thus collcting th rvious inqualitis and rcalling (7), w s from (6) J = B C D U V ( + O ( B C U ) 13/16 ( D V ) 1/2 5/18+o(1)) (8) Thus dnoting M = max { U, V } and comaring (5) with (8), w driv A B C D B C D M 2 + ( B C ) 13/16 D 1/2 M 21/16 5/18+o(1) and th rsult now follows In articular, taking A = B = C = D w s that Thorm 5 imlis that for an arbitrary st A F, w hav max{ A 3, 2A } min{ A, A 10/7 40/189+o(1) } Howvr this bound sms wakr that th on which on can driv using a combination of th bounds of Garav [13] and Rudnv [24] max{ A 3, 2A } max{ A 2, 2A } min{ A, A 2 1/2 }, max{ A 3, 2A } max{ A 2, 2A } (min{ A, }) 12/11+o(1) (9) 22 Sum Invrsion Estimats In Thorm 2 w can rlac A by A 1 and B by B 1 to asily obtain an analogu of that rsult: For arbitrary sts A, B, C F, w hav A B A 1 + C 3 { } 8 min A, A 2 B C Lmma 6 For arbitrary sts X, Y F, and a non-zro lmnt λ F, ( λ(x y) 1 ) 2 X Y x X y Y y x

6 INTEGERS: 12 (2012) 6 Thorm 7 For arbitrary sts A, B, C F, w hav A + B A 1 + C 1 { } 6 min A, A 2 B C Proof W now mimic th argumnt of Garav [13] and considr th quation c + (s b) 1 = t, (b, c, s, t) B C S T (10) whr S = A + B and T = A 1 + C For any trilt (a, b, c) A B C thr is a uniqu solution, namly t = a 1 +c, s = a + b So (10) has at last A B C solutions On th othr hand, as in [13], using Lmma 6 to stimat th total numbr of solutions to (10), on drivs A B C B C S T + 2 S B C T, which imlis th rsult (in fact with th constant 3 8 1/6) Thn as in Sction 21 w obtain: Corollary 8 For an arbitrary st A F, w hav ka ka 1 min {c A, ck 1 A 2k }, whr c = 1/6 k 1 Furthrmor, taking B = A 1 and C = A in Thorm 7, w driv: Corollary 9 For an arbitrary st A F, w hav { A, A + A 1 6 1/2 A 2 } min W also not that for smallr sts, Bourgain [3, Thorm 41] has shown that for any ε > 0 thr xists δ > 0 such for any st A F of cardinality A 1 ε, max { A + A, A 1 + A 1 } A 1+δ Th dndnc of δ on ε has not bn mad xlicit in [3], howvr using a rcnt stimat of Hlfgott & Rudnv [19, Thorm 2] or its imrovmnt du to Jons [20] in th argumnt of [3] on can asily driv such a rsult 3 Alications 31 Exonntial Congrunc For a rim and an intgr a Z with gcd(a, ) = 1 w dnot by N(; a) th numbr of solutions to th congrunc (1)

7 INTEGERS: 12 (2012) 7 By [1, Thorm 2] w hav, uniformly for t 1, N(; a) max{t, 1/2 t 1/4 } o(1), (11) a Z ord a t as, whr ord a dnots th multilicativ ordr of a F Furthrmor, for small valus of t by [1, Thorm 4] w also hav N(; a) 1/3+o(1) t 2/3, (12) a Z ord a t as W now giv an stimat that imrovs (11) and (12) for 1/4 t 2/3 Thorm 10 Uniformly ovr t 1, w hav, as, N(; a) max{t, 1/2 } o(1) a Z ord a t Proof W fix a rimitiv root g F and for u F (and so for any intgr u 0 (mod )) w us ind u for its discrt logarithm modulo, that is, th uniqu rsidu class v (mod 1) with g v u (mod ) As in th roof of [1, Thorm 2], for d ( 1)/t, w dnot by Y d th st of intgrs y satisfying th congrunc ind (dy) 0 (mod T d ), 1 y D, gcd (y, T d ) = 1, whr T d = 1 and D = 1 dt d Furthrmor, lt W d b th st of rsidu classs rrsntd by th lmnts of Y d (that is, w mbd Y d in F in a canonical way) Thn w hav N(; a) = Y d = W d ; (13) a Z ord a t d ( 1)/t d ( 1)/t s [1, Equation (9)] W not that for any intgr k 1 w hav, kw d kd and Wd k dt (which ar straight forward gnralisations of [1, Equations (10) and (11)], rsctivly, that corrsond to k = 2) Alying Corollary 3, w s that for vry fixd k } min { W d, A 2k k 1 Ddt t or W d max{t, 1/2 }t 1/2k (which substituts [1, Equation (12)]) Sinc k is arbitrary, rcalling (13), and th wll-known bound m o(1) on th numbr of intgr divisors of an intgr m 1, w driv th rsult

8 INTEGERS: 12 (2012) 8 32 Intrsctions of Almost Arithmtic and Gomtric Progrssions W say that a st I F is an almost arithmtic rogrssion if for vry fixd intgr k 1 and ral ε > 0 thr is a constant C + (k, ε) such that ki C + (k, ε) I ε W also say that a st G F is an almost gomtric rogrssion if for vry fixd intgr k 1 and ral ε > 0 thr is a constant C (k, ε) such that G k C (k, ε) G ε Thorm 11 For any almost arithmtic rogrssion I F and almost gomtric rogrssion G F w hav, ( ) I G I G + 1/2 o(1) as Proof Lt A = I G thn, for any fixd intgr k 1 w s that A k G o(1) and ka I o(1) Alying Corollary 3, w driv G I o(1) min {c A, ck 1 A 2k }, whr c = 3/8 or k 1 ( ) G I A + 1/2 ( G I /) 1/2k o(1) Sinc k is arbitrary, th rsult now follows W not that ur bounds for th numbr of rsidus modulo of conscutiv owrs g x with x [K + 1, K + M] in an intrval of lngth M that blong to som othr intrval [L + 1, L + M] of lngth M ar givn by Cillrulo & Garav [9] Th stimats and mthods of [9] imrov thos of [7] Howvr thy do not sm to aly to almost arithmtic and gomtric rogrssions (whil th aroach of [7] dos and is actually usd hr) On th othr hand, th bound (9) imlis that for M 6/11 w hav I G M 11/12+o(1) Using Corollary 8 w also driv: Thorm 12 For any almost arithmtic rogrssions I, J F w hav, ( ) I J I J 1 + 1/2 o(1) as

9 INTEGERS: 12 (2012) 9 4 Commnts In rlation to Corollary 8 it could b rlvant to rcall a wll-known xaml givn in [8], that shows thr ar infinitly many airs (, A) of rims and sts A F with A 1/2+o(1) (14) and such that for any fixd intgr k, max { A k, ka } 3/4+o(1) (15) Indd, lt H b a multilicativ subgrou of F of ordr H 3/4+o(1) (thr ar infinitly many rims for which such a subgrou xists, s [12]) By th igon-hol rincil, thr xists an s F such that if w st A = H {s, s + 1,, s + 3/4 } w hav A H 3/4 1/2+o(1) It is now asy to s that (15) holds for any intgr k On can also obtain a similar xaml limiting th ossibl growth of max{ ka, ka 1 } by considring th st J = {j 1 : j = 1,, 3/4 }, and dfining A as th most oular intrsction (in F ) of J with on of th sts {s + 1,, s + 3/4 }, s F Thrfor w s that, for an infinit numbr of rims, thr is a st A F satisfying (14) and such that, for any fixd intgr k, max{ ka, ka 1 } 3/4 Acknowldgmnts Rsarch of A B was suortd in art by Hungarian National Scinc Foundation Grants K72731 and K81658 and that of I S was suortd in art by Australia Rsarch Council Grants DP and DP Rfrncs [1] A Balog, K A Broughan and I E Sharlinski, On th numbr of solutions of xonntial congruncs, Acta Arith, 148 (2011), [2] K Bibak, Additiv combinatorics with a viw towards comutr scinc and crytograhy: An xosition, rrint, 2011, (availabl from htt://arxivorg/abs/ ) [3] J Bourgain, Mor on th sum-roduct hnomnon in rim filds and its alications, Int J Numbr Thory, 1 (2005), 1 32 [4] J Bourgain and M Z Garav, On a variant of sum-roduct stimats and xlicit xonntial sum bounds in rim filds, Math Proc Cambr Phil Soc, 146 (2008), 1 21

10 INTEGERS: 12 (2012) 10 [5] J Bourgain, A A Glibichuk and S V Konyagin, Estimats for th numbr of sums and roducts and for xonntial sums in filds of rim ordr, J Lond Math Soc, 73 (2006), [6] J Bourgain, N Katz and T Tao, A sum roduct stimat in finit filds and alications, Gom Funct Analysis, 14 (2004), [7] T H Chan and I E Sharlinski, On th concntration of oints on modular hyrbolas and xonntial curvs, Acta Arith, 142 (2010), [8] M-C Chang, Som roblms in combinatorial numbr thory, Intgrs, 8 (2008), Articl A1, 1 11 [9] J Cillrulo and M Z Garav, Concntration of oints on two and thr dimnsional modular hyrbolas and alications, Gom and Func Anal, 21 (2011), [10] E Croot and D Hart, h-fold sums from a st with fw roducts, SIAM J Discr Math, 24 (2010), [11] G Elks, On th numbr of sums and roducts, Acta Arith, 81 (1997), [12] K Ford, Th distribution of intgrs with a divisor in a givn intrval, Annals Math, 168 (2008), [13] M Z Garav, Th sum-roduct stimat for larg substs of rim filds, Proc Amr Math Soc, 136 (2008), [14] M Z Garav, Sums and roducts of sts and stimats of rational trigonomtric sums in filds of rim ordr, Uskhi Mat Nauk, 65 (4) (2010), 5 66 [15] A Glibichuk, Sums of owrs of substs or arbitrary finit filds, Izv Ross Akad Nauk Sr Mat (Transl as Izvstiya Mathmatics), 75 (2) (2011), 3 36 (in Russian) [16] A Glibichuk and M Rudnv, On additiv rortis of roduct sts in an arbitrary finit fild, J d Analys Math, 108 (2009), [17] D Hart, A Iosvich and J Solymosi, Sums and roducts in finit filds via Kloostrman sums, Intrn Math Rs Notics, 2007 (2007), Articl ID rnm007, 1 14 [18] D Hart, L Li and C-Y Shn, Fourir analysis and xanding hnomna in finit filds, Prrint, 2009 (availabl from htt://arxivorg/abs/ ) [19] H A Hlfgott and M Rudnv, An xlicit incidnc thorm in F, Mathmatika, 57 (2011) [20] T G F Jons, An imrovd incidnc bound for filds of rim ordr, Prrint, 2011 (availabl from htt://arxivorg/abs/ v1) [21] A A Karatsuba, Th distribution of valus of Dirichlt charactrs on additiv squncs, Doklady Acad Sci USSR, 319 (1991), (in Russian) [22] A A Karatsuba, Basic analytic numbr thory, Sringr-Vrlag, 1993 [23] L Li, Slightly imrovd sum-roduct stimats in filds of rim ordr, Acta Arith, 147 (2011), [24] M Rudnv, An imrovd sum-roduct inquality in filds of rim ordr, rrint, 2010 (availabl from htt://arxivorg/abs/ ) [25] C-Y Shn, An xtnsion of Bourgain and Garav s sum-roduct stimats, Acta Arith, 135 (2008), [26] I E Sharlinski, On th solvability of bilinar quations in finit filds, Glasgow Math J 50 (2008),