OPTIMAL STRONG APPROXIMATION FOR QUADRATIC FORMS

Transcription

1 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS ASER T SARDARI Abstract For a non-dgnrat intgral uadratic form F x 1,, x d in d 5 variabls, w prov an optimal strong approximation thorm Lt Ω b a fixd compact subst of th affin uadric F x 1,, x d = 1 ovr th ral numbrs Tak a small ball B of radius 0 < r < 1 insid Ω, and an intgr m Furthr assum that is a givn intgr which satisfis δ,ω r 1 m 4+δ for any δ > 0 Finally assum that an intgral vctor λ 1,, λ d mod m is givn Thn w show that thr xists an intgral solution X = x 1,, x d of F X = such that x i λ i mod m and X B, providd that all th local conditions ar satisfid W also show that 4 is th bst possibl xponnt Morovr, for a non-dgnrat intgral uadratic form in 4 variabls w prov th sam rsult if is odd and δ,ω r 1 m 6+ɛ Basd on our numrical xprimnts on th diamtr of LPS Ramanujan graphs and th xpctd suar root cancllation in a particular sum that appars in Rmark 68, w conjctur that th thorm holds for any uadratic form in 4 variabls with th optimal xponnt 4 Contnts 1 Introduction 1 Rpulsion of intgral points 7 3 Th dlta mthod 9 4 uadratic xponntial sums S m, c 11 5 Construction of th smooth wight function w 16 6 Th intgral I m, c 18 7 Th main thorm 7 Rfrncs 9 1 Introduction 11 Statmnt of rsults Bfor stating our main thorm, w discuss an application to a classical problm It addrsss th ustion of approximating ral matrics by intgral matrics Mor prcisly, lt A = [a i,j ] b a matrix of dtrminant 1 and m b a positiv intgr How wll can on approximat A by m 1 H, whr H = [h i,j ] is an intgral matrix of dtrminant m? Tijdman [Tij86], showd that thr xists H such that max m 1 hij a ij < Cm 1 18 log m 7/9, i,j Dat: August 9, 018 1

2 ASER T SARDARI whr C is a constant dpnding on max a ij Latr, Harman [Har90] improvd th xponnt 1 18 to 1 8 and showd this xponnt cannot b smallr than 1 4 Harman rmarkd that if Hooly s R Conjctur [Hoo78] wr tru, thn th xponnt drops to 1 6 Subsuntly, Chiu [Chi95, Rmark 16] rmarkd that by assuming th p-th Fourir cofficint of SL Z Maass cusp forms ar lss than p r for all prim numbrs p thn th xponnt is lss than [r + 1/ 1]/3 Thrfor, if on assums th Ramanujan Conjctur for SL, Z Maass cusp forms, thn th xponnt drops to 1 6 Th bst known bound toward th Ramanujan conjctur is 7 64 [Kim03] and this yilds , which slightly improvs Harman s bound W not that undr th most favorabl conditions, th two diffrnt mthods giv th sam xponnt 1 6 W show that th matrix approximation is possibl with th xponnt 1 6 without any assumptions Corollary 11 Fix Ω a compact subst of SL, R and any δ > 0 Thn for vry matrix A Ω and m Z, thr xists an intgral matrix H M [Z] such that dth = m and A 1 H m 1 6 +δ, m whr [a ij ] := sup a ij and th implicit constant involvd in only dpnds on δ and Ω Morovr, w cannot rplac 1 6 in th xponnt with a numbr smallr than 1 4 W first stat our main thorm in a ualitativ form Lt F X b a nondgnrat uadratic form in d 4 variabls, and Ω b a fixd compact subst of th affin uadric F X = 1 ovr th ral numbrs Thorm 1 Tak a small ball B of radius 0 < r < 1 with cntr in Ω, and an intgr m Assum that an intgral vctor λ := λ 1,, λ d mod m and an intgr ar givn such that F λ mod m Furthr assum that F X = has a local solution x p Z d p for all prims p such that x p λ mod p ordpm If d 5, assum that satisfis r 1 m 4+δ for any δ > 0, whr th constant for only dpnds on Ω and δ Thn thr xists an intgral solution F x 1,, x d = such that 11 x i λ i mod m and x 1,, x d B Morovr, th xponnt 4 in r 1 m 4+δ is optimal If d = 4 th sam rsult holds providd that is odd and r 1 m 6+δ Th xponnt 6 cannot b rplacd with a numbr smallr than 4 Basd on our numrical xprimnts on th diamtr of LPS Ramanujan graphs [Sar18, RS17, Sar17] s Rmark 15, and th xpctd suar root cancllation in a particular sum that appars in Rmark 68, w mak th following conjctur Conjctur 13 W conjctur that for d = 4 th rsult of Thorm 1 holds if is odd and r 1 m 4+δ Thrfor, th xponnt 4 is th optimal xponnt for vry uadratic form with 4 or mor variabls Rmark 14 Corollary 11 follows from Thorm 1 whn F x 1,, x 4 = x 1 x 4 x x 3 Morovr, by Conjctur 13, th matrix approximation is possibl with th optimal xponnt 1 4

3 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 3 For mor vidnc of Conjctur 13, w rfr th radr to th rcnt work of Sarnak [Sar15a, Sar15b] H provd th xistnc of th optimal lift for th cas of SL Z SL Z/Z which is shown by an lmntary mthod Anothr application of Thorm 1 rcovrs th bst known uppr bound for th diamtr of th Ramanujan graphs first constructd xplicitly by Margulis [Mar88], and latr indpndntly by Lubotzky, Phillips and Sarnak [LPS88] Mor prcisly, lt X p,m b th th LPS Ramanujan graph of dgr p + 1, and vrtics indxd by P GL Z/mZ, whr p is a prim numbr and m Z; s [LPS88, Sction ] For any x, y X p,m, lt dx, y b th lngth of th shortst path btwn x and y Dfin th diamtr of X p,m by: diamx p,m := sup x,y X p,m dx, y It is asy to s that for diamx p,m log p X p,m whr X p,m is th numbr of vrtics of X p,m In [LPS88, Thorm 51], th authors provd that diamx p,m + ɛ log p X p,m Thy provd this rsult by appaling to th Ramanujan bound on th Fourir cofficints of wight two modular forms [Eic54] W rcovr this uppr bound by applying Thorm 1 to th uadratic form F x 1,, x 4 = x 1 + 4x + 4x 3 + 4x 4; s [LPS88, Sction 3] In our proof, w us Wil s bound for th Kloostrman sums In fact, th diamtr of th LPS Ramanujan graphs was xpctd to b boundd from abov by th optimal bound 1 + ɛ log p X p,m ; s [Sar90, Chaptr 3] Howvr, Thorm 1 implis that for any prim p thr xists an infinit sunc of intgrs {m i } such that diamx p,mi 4/3 log p X p,mi ; s [Sar18, Thorm 1] Rmark 15 In [Sar18, RS17], w dmonstratd som numrical xprimnts on th diamtr of th LPS Ramanujan graphs X p,m, which shows that thir diamtr is asymptotically 4/3 log p X p,m In our xprimnts, p is fixd and m is growing This supports Conjctur 13 W now stat Thorm 1 in a strongr, uantitativ form, which provids a lowr bound on th numbr of intgral points insid an opn nighborhood of th adlic topology First, w dfin th adlic topology For vry prim p, dfin a natural norm on d p via x p = max 1 i d x i p whr x i p := p ordpxi For th archimdan plac, w fix an arbitrary norm on R d For a plac v of v = p or and a point o d v, w dfin th ball B v o, r cntrd at o with radius r by B v o, r := { x d v : x o v r } ot that for a non-archimdan plac v, th radius r taks discrt valus v n, whr v is th ordr of th rsidu fild and n Z Lt A d Z := Rd p Zd p b th intgral ring of adls W dfin a global ball insid A d Z to b a product of local ons, subjct to th condition that th radius of th local balls is 1 for all but a finit st of placs That is, 1 B a,r := B a, r p B p a p, p νp, whr a := a, a p p A d Z and r := r, p νp p R Furthrmor, w ar fr to choos a R d, a p Z d p, and ν p 0 subjct to th condition that ν p = 0 outsid a finit st of prims By this condition, m := p pνp is wll-dfind for

4 4 ASER T SARDARI som m Z W dfin th norm of th global ball B a,r to b B a,r := rm 1 Considr th following compact topological spac: Θ := Ω p V Z p, whr V R := {x R d : F x = } for any commutativ ring R Tak a global ball B a,r whr a Θ and gcda p, p = 1, i a p p = 1 Dfin th p-adic dnsity of V Z p associatd to th local ball B p a p, p νp by σ p a p, p νp, := lim k #{x Z p k+νp Z d : F x = mod p k+ν p and x a p mod p νp } p kd 1 If ν p = 0 which mans w don t hav any mod p congrunc condition thn w simply writ σ p It follows from Hnsl s lmma that th abov limit xists Th condition σ p a p, p νp, 0 is uivalnt to V Z p B p a p, p νp Lt S Ba,r := p σ pa p, p νp, b th singular sris If ν p = 0 for all prims p, thn w writ S := p σ p for th singular sris For d 4 th product dfinition of S Ba,r is absolutly convrgnt and th singular sris is nonzro if and only if σ p a p, p νp, 0 for vry p; s [Si67] Hnc, if S Ba,r 0 thn V Z p B p a p, p νp for vry prim p, and w say that all th local conditions ar satisfid Th uantitativ form of our main thorm is as follows Thorm 16 If d 5, thn w hav 13 V Z B a,r S Ba,r B a,r d 1 d 1 + O B a,r d 3 ɛ for som ɛ > 0, whr th implid constant in and O only dpnds on ɛ and Ω and not on or th global ball B a,r For d = 4, w hav 14 V Z B a,r S Ba,r B a,r O B a,r 3 1 ɛ Rmark 17 If d 5, thn w hav [Mal6, Rmark 7 Pag 73] 15 c ɛ ɛ S Ba,r, whr ɛ > 0 is any positiv ral numbr and c ɛ only dpnds on Ω and ɛ Thrfor, if δ B a,r for som δ > 0 thn th rror trm is smallr than th main trm, and as a rsult w hav an intgral point insid B a,r which implis Thorm 1 for d 5 For d = 4, th inuality 15 holds if is odd Similarly, if δ B a,r 3 thn w hav an intgral point insid B a,r and this implis Throm 1 for d = 4 Morrovr, if gcdp νp, a 1 p,, a d p = p hp 1 for som p, thn w can divid by p hp, rplac th local balls at th plac p by B ap, p hp p νp hp, and us th abov thorm Sinc th powr of is d and th powr of B a,r is d 1 in th formula 13, th main trm would b multiplid by p php W now stat a thorm which implis that th xponnt 4 in Thorm 1 is optimal It is basd on th principl that rational points on V 1 of low hight rpl othr rational points Lt := 1,, d b any primitiv intgral vctor, i such that gcd 1,, d = 1, with F > 0 Lt ν := ν p b a sunc of nonngativ intgrs indxd by all prims p such that ν p = 0 for all but a finit st of prims, and if ν p 0 thn = whr is th Lgndr symbol p F p

5 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 5 Morovr, lt a := a p b a sunc of p-adic intgrs a p Z p indxd by all prim numbrs such that if ν p 0, thn a p is on of th two solutions of a p = F, and if ν p = 0 thn a p = 0 Dfin 16 B,ν,a := B a, r p whr a = 1 Our rsult is th following F B p a p, p νp, Thorm 18 Thr xist positiv numbrs c 1 and c dpnding only on Ω and F X such that whnvr on has 17 B,ν,a c /, thn thr xists a global ball B insid B,ν,a with no intgral points and a norm gratr than B > c 1 4 1/ In th last application of our main thorm, w giv a lowr bound for th numbr of lattic points insid a small cap on a sphr, i th intrsction of a ball with a small radius and th sphr In th appndix of [BR1], an uppr bound for th numbr of lattic points in such small caps is givn Following a suggstion of Bourgain [Sar15a, Pag 3 ], w promot this to a lowr bound Corollary 19 Lt S d 1 R b any d 1-dimnsional sphr of radius R cntrd at th origin such that := R is an intgr and d 5 Suppos w ar givn a sphrical cap of diamtr Y R Thn, th numbr of intgral lattic points insid this sphrical cap is at last σ S Y d O R ɛ R R Y d 3, whr σ is a constant that only dpnds on d and S is th singular sris associatd to th uadric x x d = Th implid constant in O dpnds on ɛ and d but not on th sphrical cap or In particular, if Y δ R 1/+δ for any δ > 0 thn w hav an intgral point insid th cap For d = 4, if w assum that is odd thn th numbr of intgral points is at last σ S Y O R ɛ R R Y 3/ In this cas, if Y δ R /3+δ thn w hav an intgral point insid th cap On th othr hand, it follows from Thorm 18 that thr ar caps of diamtr Y R 1/ on any S d 1 R with no intgral points insid thm, whr th implid constant in only dpnds on d Givn R > 0 such that R Z, w lt CR dnot th maximum volum of any cap on S d 1 R which contains no intgral points Sarnak dfind [Sar15a] th covring xponnt of intgral points on th sphr by: K d := lim sup R log #S d 1 R Z d log vol S d 1 R/CR It follows from Thorm 18 and Corollary 19 that K d = d 1 4/3 K 4 for d 5 and

6 6 ASER T SARDARI Rmark 110 In his lttr [Sar15a] to Aaronson and Pollington, Sarnak showd that 4/3 K 4 To show that K 4 h appald to th Ramanujan bound on th Fourir cofficints of wight k modular forms, whil th lowr bound 4/3 K 4 is a consunc of an lmntary numbr thory argumnt Furthrmor, Sarnak stats som opn problms [Sar15a, Pag 4] Th first on is to show that K 4 < or vn that K 4 = 4/3 Conjctur 13 implis that K 4 = 4/3 Mor gnrally, Ghosh, Gorodnick and vo studid th covring xponnt of th orbits of a lattic subgroup Γ in a connctd Li group G, acting on a suitabl homognous spacs G/H; s [GG13, GG15, GG16] Thy linkd th covring xponnt of Γ to th spctrum of H in th automorphic rprsntation on L Γ\G In particular, thy showd that K Γ if th rstriction of th unitary rprsntation on L Γ\G to H has tmprd sphrical spctrum as a rprsntation of H; s [GG16, Thorm 35] and [GG15, Thorm 33] This rcovrs th abov rsult of Sarnak for d = 4 For d 5, by using th bst bound on th gnralizd Ramanujan conjctur for SO d, thy showd that [GG13, Pag 1] 1 K d 4 4/d 1 for odd d and 1 K d 4 16/d + for vn d Thy raisd th ustion of improving ths bounds in [GG13, Pag 11] As pointd out abov, w giv a dfinit answr to this ustion and show that K d = d 1 for d 5 1 Furthr motivations and tchnius In svral paprs, Wright provd various rsults about th rprsntation of an intgr as a sum of suars of intgrs almost proportional to assignd positiv ral numbrs λ 1, λ,, λ d In [Wri33], h showd that if ɛ U for som 0 < ɛ and λ λ 5 = 1, whr 0 < λ i, thn thr xists an intgral solution n 1,, n 5 to = n n 5, whr n i λ i < U ot that by Thorm 1 th inuality ɛ U is not sharp and can b improvd to ɛ U H also showd that th numbr of rprsntation is U 4 By an ntirly lmntary mthod Auluck and Chowla in [AC37] provd a sharp rsult for th spcial point λ 1,, λ 4 = 1 4, 1 4, 1 4, 1 4 and sum of four suars Thy showd that vry positiv intgr 0 mod 8 is xprssibl in th form = n 1 + +n 4, whr n i ar intgrs satisfying 4 n i = O 3 4 It follows from furthr work of Wright [Wri37] that O 3 4 in this thorm cannot b rplacd by o 3 4 Mor gnrally, h considrd th sum of d kth powrs and provd that thr xists an infinit sunc of intgrs { i } such that th diagonal point W d,k,i := a i,, a i, whr da k i = i for som a i R, rpls th intgral points on x k x k d = i Mor prcisly, h showd that th ball B Wd,k,i,γ d,k 1/k i cntrd at W d,k,i with radius γ d,k 1/k i for som fixd γ d,k > 0 contains no intgral points x 1,, x d such that x k x k d = i W will discuss this rpulsion proprty for k = in mor dtail in Sction In a rcnt papr [Da10], Damn gav a lowr bound for th numbr of intgral points x 1,, x d clos to th diagonal point W d,k, such that x k x k d = H provd that for vry k thr xists d k such that if d d k, thn on has a lowr bound for th numbr of intgral points insid B Wd,k,i for any ɛ > 0 and larg nough This, 1/k+ɛ lowr bound diffrs by a boundd scalar compard to th uppr For th sum of suars, his rsult implis that d 9 H rmarkd that by working a littl hardr, on can prov that d 7 Thorm 1 implis that d 5

7 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 7 W shall prov Thorm 1 using a vrsion of th dlta mthod, which in turn is basd on th Kloostrman circl mthod This mthod was dvlopd by Kloostrman to prov th local-to-global principl for uadratic forms in 4 variabls; s [Klo7] For th purpos of rprsnting intgrs by uadratic forms, if th numbr of variabls is 5 or mor th classical circl mthod of major and minor arcs works fin Howvr, for 4 variabls it dos not work Kloostrman introducd a nw mthod of dissction by th Fary sunc with no minor arcs which dals with 4 variabls In our work, th mthod provs to b dcisiv in 5 or mor variabls as it givs th optimal xponnt for lifting solutions Our optimal rsult for 5 variabls dpnds crucially on obtaining suar root cancllation in crtain xponntial sums, and also rlis on a rfinmnt of th Kloostrman mthod dvlopd by Duk, Fridlandr and Iwanic known as th dlta mthod; s [DFI93] In th dlta mthod w us a smooth cut off function ovr th Fary dissctions Th dlta mthod allows us to us th rapid dcay of th Fourir transform of th wight function and maks computations handir For proving Thorm 1, w us a vrsion of this mthod dvlopd by Hath-Brown in [HB96] In that mthod w first apply th dlta mthod and thn a Poisson summation ovr th sum of lattic points Our main innovation is to introduc a spcial coordinat systm in Sction 5 which is crucial in improving th currnt bounds on th oscillatory intgrals apparing in th dlta mthod Th problm of th distribution of intgral points on uadrics btwn diffrnt rsidu classs to a fixd intgr m has bn studid by Malyshv; s [Mal6] Malyshv usd Kloostrman mthod and provd a rsult about th distribution of intgral points of th uadric F X = with 4 or mor variabls btwn rsidu class of an intgr m An application of Thorm 1 significantly improvs th xponnts of m and in his main thorm for F X fixd Acknowldgmnts I would lik to thank my PhD advisor Ptr Sarnak for svral insightful and inspiring convrsations during th cours of this work In fact th starting point of this work was his lttr [Sar15a] and in particular th ky obsrvation of Bourgain notd thr as wll as abov I would lik to thank Profssor Jianya Liu and Profssor Tim Browning for thir commnts on th arlir vrsions of this work Finally, I am gratful for th commnts of Simon Marshall, Masoud Zargar, and th anonymous rfrs that improvd th writing of this manuscript Rpulsion of intgral points In this sction w prov Thorm 18 W assum that F X = X T AX is a non-dgnrat uadratic form Rcall th dfinition of B,ν,a in 16, and th notation usd whil formulating Thorm 18 Proof Lt B,ν,a := p B pa p, p νp b th finit part of th global ball B,ν,a, and m := p pνp W assum that X V Z is an intgral point such that X B,ν,a Hnc, X a p mod p ordpm By th Chins rmaindr thorm thr xists α Z such that X α mod m, whr α a p mod p ordpm Hnc, X = mt + α for som intgral vctor t W writ th Taylor xpansion of F mt + α at α, i F mt + α = m F t + mt T Aα + F α = Sinc gcdα, m = 1, w dduc that t T A F α mα mod m Thrfor, thr xists a fixd numbr

8 8 ASER T SARDARI l 0 mod m such that 1 X, A mt + α, A l 0 mod m ow considr th infinit part of th global ball B,ν,a, namly B := B a, r Without loss of gnrality w assum that B intrscts only on connctd componnt of V 1 R W wish to find a ral point B V R such that, A = m + l 0 + km for som k Z, and a is minimal W will dduc th xistnc of from assumption 17 although th w produc is not ncssarily uniu By th connctivity of B V R and th continuity of th innr product, it suffics to show that th lngth of th intrval I := { h, A R : h B V R} is biggr than m For any r < 1 and y Ω, lt J y,r := { l, Ay : l V 1 R, l y < r} R Sinc Ω is compact and F X is non-dgnrat, it follows that thr xists a constant c 3 dpnding on Ω such that th lngth of J y,r is biggr than c 3 r, indpndntly of y Ω Morovr, thr xist constants c 4 and c 5 dpnding on Ω such that c 4 F c 5 This implis that J a,r c 3 r, or uivalntly that I c 3 r F c 3 c 4 r By assumption 17, w hav m c 1 r By choosing c 1 < c 3 c 4, it follows that I m, and hnc thr xists satisfying condition xt, w giv an uppr bound on a By condition and th connctivity of B V R, it follows that a, A m W writ a = 1 +, whr 1 is paralll to A and is orthogonal to A W hav a, A 1 = 1, A A Sinc F = F a =, w hav = A m A m Ω T A = T 1 A A a 1 + m a Hnc, 1 + m 1/4 a 1/ By th triangl inuality and th assumption 17, 3 a = 1 + m + m 1/4 a 1/ m 1/4 a 1/ Assum that X = + ξ is an intgral point insid B,ν,a with F X = W writ th Taylor xpansion of F X at and obtain F X = F + T Aξ +F ξ Sinc F X = F =, w dduc that T Aξ + F ξ = 0 Hnc, F ξ + ξ, A a By 1 and th dfinition of in, ξ, a A ξ, a A m a By th Cauchy-Schwarz inuality and inuality 3, ξ, A a ξ A a m 1/4 a 1/ ξ Thrfor, F ξ + m 1/4 a 1/ ξ m a

9 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 9 W dduc that ξ ma 1/ 1/4 Sinc a = 1/ F 1, thn thr xists a constant c dpnding on Ω and F such that ξ m c 1/4 Lt r m := c 1/4 1/ 1/ and B := B, r p Ba p, p ordpm Thn, B m = c 1/4 and B dos 1/ not contain any intgral point This concluds th proof of Thorm 18 3 Th dlta mthod W now bgin th proof of Thorm 16 In this sction, w dfin a smooth sum w, λ that givs a lowr bound for th numbr of intgral points w wish to study, and us th dlta mthod to giv an xpansion for w, λ Assum that F X = X T AX is a non-dgnrat uadratic form Rcall th notations usd whil formulating Thorm 16 for F X Assum that X := x 1,, x d is an intgral point insid th givn global ball B a,r Hnc, 31 x i a i p mod p νp, for vry p, whr a p = a 1 p,, a d p Z d p Rcall that B a,r = m 1 r whr m := p pνp By th Chins rmaindr thorm thr xists λ = λ 1,, λ d mod m such that 3 x i λ i mod m, if and only if th systm of congrunc conditions 31 hold So, counting intgral points insid B a,r is th sam as counting intgral points insid th ball B a, r R d subjctd to th congrunc condition 3 xt, w giv an xpansion of th dlta function which is dvlopd by Duk, Fridlandr and Iwanic [DFI93] W cit this thorm from [HB96, Thorm 1] Thorm 31 For any intgr n lt { 1 if n = 0, δn = 0 othrwis Thn for any > 1 thr is a positiv constant C, and a smooth function hx, y dfind on th st 0, R, such that 33 δn = C an h, n, =1 a whr mans th sum is ovr a mod with gcda, = 1, and th constant C satisfis C = 1 + O, for any > 0 Morovr hx, y x 1 for all y, and hx, y is non-zro only for x max1, y Lt w b a smooth compactly supportd wight function dfind on R n such that 34 wx = 0 if x / B a, r Assum that X Z d satisfis th condition 3 W uniuly writ X = mt + λ, whr t Z d and λ = λ 1,, λ d for m < λ i m Dfin 35 k := F λ m Sinc F X =, thn m F t + mλ T At = F λ which implis m λ T At k Thn, F t + 1 m λt At k = 0 W also dfin Gt := F mt + λ m = F t + 1 m λt At k

10 10 ASER T SARDARI Finally, w dfin w, λ := t wmt + λδgt, whr t Z d ot that w, λ is th wightd numbr of intgral points X satisfying condition 3 W wish to apply th dlta xpansion in 33 to δgt ot that 33 holds only for n Z Morovr, Gt Z if and only if m λ T At k Hnc, w writ w, λ = 1 m l m λt At k wmt + λδgt, l whr l varis mod m Thn, w apply 33 with := mr 1 w, λ = C m l t a t and obtain l + aλ T At k + amf t m h, Gt wmt + λ W apply th Poisson summation formula on th t variabl to obtain 36 w, λ = C m l a c m d S m, a, l, ci m, c, whr c Z d, l Z/mZ, and a Z/Z, and I m, and S m, ar givn by 37 I m, c := h, Gt wmt + λ c, t dt1 dt d, m 38 S m, a, l, c := b Z/mZ d l + aλ T Ab k + amf b + c, b m W dfin th sum S m, c as: Sm, a, l, c 39 S m, c := l Thrfor 310 w, λ = C m m d S m, ci m, c c Sinc = mr 1, in th Sction 5, w construct a smooth function w with compact support such that 311 wmt + λ 0 only if Gt 1 Sinc th support of th smooth function hx, y is insid th intrval 0 x maxy, 1 Hnc, th summation on in formula 310 is rstrictd to 1 a

11 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS Th proprtis of th smooth function hx, y In this sction w cit th basic proprtis of th smooth function hx, y from [HB96] Lmma 3 W hav for any l 0 and k {0, 1} x k y l k hx, y x k dy l x l, Proof This is an asy consunc of [HB96, Lmma 5] Th following lmma shows hx, y convrgs to th dlta function rapidly as x 0 For th proof w rfr th radr to [HB96, Lmma 9] Lmma 33 Lt f b a smooth function with compact support Thn if x 1 w hav 31 fyhx, ydy = f0 + O M,f x M Th following lmma shows th smooth proprty of th hx, y in th y variabl Sinc it convrgs to th dlta function from th prvious lmma, as x 0 th dcay rat of th Fourir transform is slowr as x 0 For th proof, w rfr th radr to [HB96, Lmma 17] Lmma 34 Lt w b a smooth compactly supportd wight function and pt, x b th Fourir transform of wyhx, y in th y variabl, i, pt, x = wyhx, y tydy Thn pt, x dcays fastr than any polynomial in th variabl xt, i for vry 0 w hav pt, x w, xt 4 uadratic xponntial sums S m, c Rcall th dfinition of S m, c from 39 In this sction, w prov th following uppr bound on th avrag norm of S m, c with rspct to A vrsion of this inuality was provd in Hath-Brown s papr [HB96, Lmma 8] Our proof uss Wil s bound rsp Salié s bound on gnralizd Kloostrmann sums rsp Salié sum for vn rsp odd dimnsions Proposition 41 W hav th following uppr bound X =1 whr X = O A for any fixd powr A m d d+1 Sm, c m ɛ X 1+ɛ, W giv th proof of Proposition 41 at th nd of this sction W bgin by proving som auxiliary lmmas Lmma 4 Unlss c l + aaλ mod m, w hav S m, a, l, c = 0 As a rsult S m, c = 0, unlss c αaλ for som scalar α mod m

12 1 ASER T SARDARI Proof W writ th vctor b in th sums S m, a, l, c as b = b 1 + b, whr b 1 is a vctor mod m and b is a vctor mod Thn w writ S m, a, l, c as a summation ovr b 1 and b, and obtain S m, a, l, c = l + aλ T Ab k + amf b + c, b m b It is asy to chck that th summation ovr b 1 is zro, unlss c l + aaλ mod m Hnc, w conclud th lmma In th rst of this sction w giv an uppr bound on S m, c Lt := dt A By 39 S m, c = l + aλ T Ab k + amf b + c, b m l a b Sinc th summation ovr l is nonzro only if m λ T Ab k and it is m whn m λ T Ab k, w hav 41 S m, c = m aλ T Ab k + amf b + c, b, m b,a whr th summation is ovr a Z/Z and vctors b Z/mZ d such that m λ T Ab k W assum that = 1, whr gcd 1, m = 1 and th st of prims which divid is a subst of prim divisors of m So, gcd 1, m = 1 By th Chins rmaindr thorm w writ 4 k = m k k for som intgrs k 1 and k, and a = a a, whr a 1 Z/ 1 Z and a Z/ Z W also writ b = m b b, whr b 1 Z/ 1 Z d and b Z/m Z d such that m λ T Ab k W substitut k = m k k, a = a a, and b = m b b in formula 41 for S m, c, and obtain 43 S m, c = S 1 S, whr 44 S 1 := and a 1,b 1 45 S := m a 1 λ T Ab 1 + a 1 m F b 1 + c, b 1 a 1 k 1, 1 a,b 1 a λ T Ab + a m1f b + c, b 1 a k m In th following lmma w giv uppr bounds for S 1 This lmma and its argumnt is similar to [HB96, Lmma 6] b 1 l + aλ T Ab 1 + c, b 1 m

13 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 13 Lmma 43 Lt GX := X T BX and D := dtb 0, whr B is a symmtric d d intgral matrix Lt b an intgr such that gcd, D = 1 For t Z/Z and c, c Z/Z d, w dfin SG, c, c, t := agb + c, b + t + c, b, a,b whr th sum bing takn ovr all a Z/Z d and b Z/Z Thn D 46 SG, c, c, t = τ d KlG, c, c, t, whr τ := x x is th Gauss sum, is th Jacobi symbol, and KlG, c, c, t is ithr a Kloostrman sum or a Salié sum As a rsult, w hav S 1 d+1 1 τ 1 gcd 1, 1/ Rmark 44 W not that th xponnt d+1 in th abov uppr bound is optimal, and corrsponds to th full suar root cancllation in both th a and b variabls Proof Sinc is odd w can diagonaliz our uadratic form GX mod, and writ GX = d i=1 α ix i Hnc, SG, c, c, t := a at d aαj b + c j b + c jb j=1 whr b Z/Z W complt th suar to obtain SG, c, c, t : = a W not that whr τ := x x Hnc, SG, c, c, t = τ d = a D b at at b ac d aα j b + j +cj aα j j=1 d j=1 b ac j +cj 4aα j ac aα j b + j +cj aα j = b, ac j +cj 4aα j ac aα j b + j +cj aαj τ, is a uadratic Gauss sum and aαj c jc j j α j a a xt, w analyz th sum ovr a W dfin c jc j KlG, c, c j α, t := j a a d at j d at j aα j is th Jacobi symbol c j 4α j a 1 j c j c j 4α j a 1 j c j 4α j 4α j

14 14 ASER T SARDARI d a ot that = a for odd d So, KlG, c, c, t is a Salié sum for odd d and from standard bounds on Salié sums w hav 47 KlG, c, c, t τ d a Similarly, = 1 for vn d So, KlG, c, c, t is a Kloostrman sum for vn d and by Wil s bound on Kloostrman sums w obtain 48 KlG, c, c, t τ gcd, t c j, c j 1/ 4α j j 4α j j This concluds th first part of our lmma Finally, w analyz S 1 W not that by a chang of variabls S 1 = SG, c, c, t, whr G = m F, t = k 1, and c = Aλ Rcall that F X = X T AX, gcd 1, m D = 1, and G = m F is diagonalizabl with ignvalus {α i }, so that W substitut c = Aλ and obtain 49 j c j 4α j m c T A 1 c mod 1 j c j λt Aλ 4α j m F λ m mod 1 W apply formula 46 and obtain S 1 = d/ 1 KlG, c, c, t If d is odd thn by inuality 47 w obtain S 1 d+1 1 τ 1 If d is vn thn by inuality 48 w obtain S 1 d+1 1 τ 1 gcd 1, t c j j 4α j 1/ W substitut t = k 1 and by 49 obtain S 1 d+1 1 τ 1 gcd 1, k 1 + F λ 1/ m By 4 and 35, w hav gcd 1, k 1 + F λ m = gcd 1, Hnc, This concluds our lmma S 1 d+1 1 τ 1 gcd 1, 1/ In this lmma w giv an uppr bound on S This lmma is a variant of [HB96, Lmma 5] and its proof follows from th standard suar root canclation in th Gaussian sums Lmma 45 W hav S m d 1+d/ Proof By th Cauchy-Schwarz inuality on th a variabl, w hav 410 S m φ a b = m φ a b,b 1 a λ T Ab + a m 1F b + c, b 1 a k m 1 a λ T Ab b + a m1f b F b + c, b b, m

15 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 15 whr m λ T Ab k and m λ T Ab k W chang th variabls and writ u = b b Hnc, S m φ 1 a λ T Au + a m1b Au + F u + c, u, m a u,b whr m λ T Ab k and m λ T Au Th summation ovr b is zro unlss gcdu In othr words, th summation is non-zro only if Sinc b Z/m Z d, thn u Z/ gcd, m Z d Z/ gcd, mz d S m φ a u Z/ gcd, mz = m φ d m d d, whr m λ T Ab k and m λ T Au Hnc, d b Z/m Z d 1 This concluds th lmma S m d 1+ d Rmark 46 W not that th xponnt 1+ d coms from th suar root cancllation in th b variabl, and aftr th Cauchy-Schwarz inuality in th first lin of th proof w los all th potntial cancllation in th a variabl Mor prcisly, th sum ovr a in th scond lin of th proof is ovr th suar norm of som d-dimnsional Gauss sums ovr b It follows from our proof that for fixd a ach Gauss sum is at last m d 1 d/ up to a constant which only dpnds on th discriminant This implis our uppr bound in Lmma 45 is sharp aftr th Cauchy-Schwarz inuality in th scond lin of th proof Proof of Proposition 41 This is a consunc of Lmma 43 and Lmma 45 W factor = 1, whr gcd 1, m = 1 and th prim factors of ar a subst of th prim factors of m From th formula 43, w dduc that S m, c = S 1 S By Lmma 43 and Lmma 45, w hav X =1 m d d+1 Sm, c X =1 By th Cauchy-Schwarz inuality, X =1 1/ ɛ 1 1/ gcd 1, 1/ X ɛ X =1 gcd 1, 1/ X 1/ X gcd 1, 1/ =1 =1 1/ gcd 1, 1/ It is asy to chck that X =1 gcd 1, X 1+ɛ For th othr trm, w hav X =1 d m,d<x d X d X Thrfor, w conclud th proposition d m,d<x 1 Xm X ɛ

16 16 ASER T SARDARI 5 Construction of th smooth wight function w In this sction w construct w Cc R n that w us in th dlta mthod W dfin w in an appropriat coordinat systm, which maks th computations asir for th oscillatory intgrals I m, c Finally w giv uppr bounds on th partial drivativs of w, that w us in th nxt sction W introduc th coordinat systm first Lt 51 v := a V R, and lt d b th norm 1 vctor in th dirction of F v th gradint of F at v that is Av Lt T v V R b th tangnt spac of V R at v, which is th orthogonal complmnt of d W rstrict th uadratic from F x to th d 1- dimnsional subspac T v V R By a standard thorm in Linar Algbra, w can find an orthogonal basis B 1 := { 1,, d 1 } for T v V R, such that F d 1 d 1 u i i = µ i u i i=1 xt, w construct th smooth wight function w that satisfis th ruird conditions 34 and 311 By ths conditions, it follows that th smooth wight function w is supportd insid a cylindr cntrd at v with hight r and radius r, such that its bas is paralll to T v V R Mor prcisly, lt i=1 yx := F x m Sinc F v = v T Av = 0 and T i Av = 0 for 1 i d 1, thn 5 B := { 1,, d 1, v} and B 3 := B 1 { d } ar bass for R d Givn x R d, w xprss th vctor x v in basis B 3 as x v = d 1 i=1 mu i i + mα d, and in B as follows: d 1 53 x v = mũ i i + mβv i=1 If v = d i=1 v i i thn u i = ũ i + βv i, and α = βv d Suppos ψ 1 Cc R and ψ Cc R d 1 W dfin th smooth wight function wx as follows: 54 wx = whr ũ := ũ 1,, ũ d 1 { x T Av ψ v,d 1y ψ ũ if 1 + mβ > 1/, 0 othrwis, x Rmark 51 Th factor T Av v,d is in our wight function w in ordr to simplify th oscillatory intgral I m, c in Lmma 6 ot that th support of th wight function ψ 1 y ψ ũ is localizd around v and v W dfin w such that it only has a support nar v In Lmma 5, w chck that w satisfis th ruird conditions 34 and 311 Morovr, w giv uppr bounds on its partial drivativs

17 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 17 Lmma 5 Writ w = w 0 ψ 1 whr w 0 x := ψ v,d 1yψ ũ Considr th xt Av coordinat systm x 1,, x d whr x i = ũ i / and x d = β/m Thn w 0 x = 0 if max x i > C, 1 i d whr C is a constant that only dpnds on Ω, A and th support of ψ 1 and ψ Morovr, for vry 1 i d and n 0, w hav n w 0 x n < C n, i whr th constant C n only dpnds on A, compact st Ω, and max d j ψ 1 j dy and j max j ψ i,j x whr 0 j n j i Proof Sinc ψ 1 Cc R and ψ Cc R d 1, thr xists a constant C, such that if max 1 i d 1 x i > C or y > C thn w 0 x = 0 Assuming that w 0 x 0, thn 1 + mβ > 1/, y C and x i < C for 1 i d 1 W xprss y in trms of x 1,, x d 1, x d mũi i mβv T A mũi i mβv y = m d 1 1 ũ i m µ i mβ 55 = m d 1 = x m i µ i + x d + x d 1 m Thrfor, x d + x d C + d 1 1 C µ i By 54, w hav m + x d = + mβ > 3/ Thn x d < C + d 1 1 C µ i W dfin C := C + d 1 1 C µ i This concluds th first part of our lmma xt, w giv uppr bounds on th partial drivativs of w W assum that x i < C for 1 i d W apply Libniz s product rul and obtain n w 0 x n i = j 1+j +j 3=n n j 1, j, j 3 j 1 x T Av v,d x j1 i j ψ 1 y j3 ψ x 1,, x d 1 x j i x j3 i Hnc, it suffics to show that th partial drivativs of ach factor on th right hand sid is O1 By 53, x T Av v, d = vt Av v, d + x vt Av v, d = vt d 1 Av v, d + T i mx Av i v, d + m v T Av x d / v, d i=1 Sinc v T Av = and T i Av = 0 for 1 i d 1, x T Av v, d = v, d + m v, d x d

18 18 ASER T SARDARI W hav x d < C in th support of w Sinc = m r, m v, d v, d = a T A a = O1 By th chain rul and Libniz s product rul, w writ j ψ 1y x j i in trms of j 1 ψ 1 y j 1 and j y, whr 1 j 1, j j Hnc, it suffics to show that y = O1 By x j i 55, y is a uadratic form in x 1,, x d 1, x d Consuntly, it suffics to show that th cofficints µ i and m / ar O1 ot that th µ i ar boundd by A th norm of A and m / < 1 Thrfor, w conclud our lmma 6 Th intgral I m, c In this sction w giv uppr bounds on th intgral I m, c W us th sam notations as in Sction 5 W partition th intgral vctors c 0 Z d into two sts, and dnot thm by ordinary and xcptional vctors In what follows, w dscrib thm Rcall th orthonormal basis B 3 dfind in 5 W writ a givn intgral vctor c in this basis as 61 c = d c i i i=1 W rmark that c dos not ncssarily hav intgral coordinats anymor W dfin two typs of ordinary vctors Th typ I ordinary vctors c ar th intgral vctors c, such that 6 c mr 1 1+ɛ, whr c = max i c i Th typ II ordinary vctors c ar th intgral vctors c, such that max c i mmr 1 ɛ, 1 i d 1 c < mr 1 1+ɛ W call th complmnt of ths intgral vctors th xcptional intgral vctors Rcall that x = mt + λ whr t = t 1,, t d and lt dt = dt 1 dt d W rcord som formulas for th volum form dt in diffrnt coordinat systms Rcall B from 5 and its coordinat systm ũ 1,, ũ d 1, β W hav 63 dt = v, d dũ 1 dũ d 1 dβ xt, w giv a formula for dt in trms of dũ 1 dũ d 1 dy y Hnc, β = xt Av m 64 dt = v, d dũ 1 dũ d 1 dβ = m x T Av dũ 1 dũ d 1 dy x j i By 55, w hav Finally, w chang th variabls as in lmma 5 to x i = ũ i / for 1 i d 1 and x d = β/m 65 dt = md+1 v, d x T Av dx 1 dx d 1 dy = md+1 v, d dx 1 dx d 1 dx d

19 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS Uppr bound on I m, c for th ordinary vctors c In this sction w assum that c is an ordinary vctor, and prov that I m, c rapidly dcays as c Morovr, w show that th contribution of th ordinary vctors c in th dlta mthod is boundd Part of our analysis in this sction is inspird by th work of Hath-Brown In particular, th us of Fourir invrsion in th proof of th following lmma is similar to [HB96, Lmma 17] Lmma 61 Lt c b an ordinary typ I vctor Thn for any k > 0, w hav 66 I m, c k c r 1 m k On th othr hand, if c is an ordinary typ II vctor, w hav 67 I m, c k r 1 m k Morovr, C m d m S m, ci m, c 1, c whr is th sum ovr all ordinary vctors and th implid constant in only dpnds on ɛ in th dfinition of th ordinary vctors, Ω, and th fixd functions ψ 1 and ψ Proof Rcall from 37 I m, c = h, Gt wmt + λ c, t dt m By 54, wmt + λ = w 0 mt + λψ 1 y, whr y = F mt+λ m b th Fourir transform of h, yψ 1y, w hav Thrfor, p ξ := ψ 1 yh, y ξydy ξ I m, c = p ξ w 0 mt + λ ξy t t, c dtdξ m = Gt Lt p ξ W chang th coordinat systm to ũ 1,, ũ d 1, β and apply formula 63 to obtain 68 I m, c = v, d p ξ w 0 ũ, β t, c ξy dũdβdξ, m ξ whr dũ = dũ 1 dũ d 1 By 55, w hav d 1 1 ũ i m µ i mβ y = m t W also writ th innr product t, c in trms of ũ 1,, ũ d 1, β, and obtain v λ 69 t, c = m x v, c + m, c d 1 = t 0, c + i=1 c i ũ i + β c, v,

20 0 ASER T SARDARI whr v = mt 0 + λ W substitut th abov xprssions in uation 68, and obtain I m, c = c, t 0 v, d p m ξ ξ w 0 ũ 1,, ũ d 1, β ξβ/m + β β v, c β,ũ m d 1 [ ξµ iũ i i=1 c iũ i m ] dβdũdξ W ar only concrnd with I m, c, so w drop c,t0 m W chang th variabls to x i = ũ i / and x d = β/m By uation 65, w obtain I m, c = d+1 m v, d p ξ w 0 x ξ x d ξ + v, c + ξx d m d 1 [ξµ i x i + c ix i m ] dx 1 dx d dξ i=1 At this point, w assum that c is an ordinary vctor of typ II, which mans thr xist 1 i d 1 such that c i mmr 1 ɛ W writ I m, c = I m, c 1 + I m, c, whr I m, c 1 is th intgral ovr ξ < r 1 m ɛ/, and I m,c is th intgral ovr ξ > r 1 m ɛ/ First, w show that 610 I m, c r 1 m k Lmma 34 implis that p ξ ξ k Sinc ξ > r 1 m ɛ/, w dduc th claimd uppr bound on I m, c It rmains to show that I m, c 1 r 1 m k W first tak th intgration on th x i variabl and obtain I m, c = d+1 m v, d p ξ ξ x d ξ + v, c + ξx d m d 1 [ξµ i x i + c ix i m ] dx 1 dx ˆ i dx d dξ i=1 w 0 x 0,, x i,, x d ξµ i x i + c ix i dxi m By Lmma 5, th wight function w 0 x 0,, x i,, x d has compact support in a fixd intrval [ C, C], and its partial drivativs ar boundd n w 0 x < C n n Sinc i ξ < r 1 m ɛ/ and c i mmr 1 ɛ, w dduc th following lowr bound on th drivativ of th phas function in th x i variabl ξµ i x i + cixi m c ix i mr 1 ɛ mr 1 ɛ x i m

21 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 1 By intgrating by parts multipl tims in th x i variabl, w hav wx i ξµ i x i + c ix i dxi mr 1 kɛ, x i m for vry k > 0 Hnc, if c is a typ II vctor, w provd th claimd uppr bound on I m, c 1, and so on I m, c xt, w show that th contribution of th typ II vctors is boundd By Proposition 41 and 67, C m m d S m, ci m, c max c I m, c m =1 c r 1 m m k m d S m, c =1 c 1 d +ɛ m ɛ c whr c varis ovr typ II vctors Sinc c r 1 m 1+ɛ, th numbr of typ II vctors is boundd by r 1 m d1+ɛ Thrfor, by choosing k larg nough, w conclud our Lmma for typ II vctors It rmains to prov th lmma for typ I vctors c W writ I m, c = I m, c 1 + I m, c, whr I m, c 1 is th intgral ovr ξ < c r 1 m ɛ/ and I m,c is th intgral ovr ξ > c r 1 m ɛ/ not that th dfinition of I m,c 1 and I m, c for th typ I ordinary vctors ar slightly diffrnt from th analogous ons for th typ II ordinary vctors From th sam lins as in 610, w hav I m, c c r 1 m k It rmains to prov I m, c 1 c r 1 m k for th typ I vctors c If c d max i c i, w procd as bfor So, w assum th hardr cas whr c d = c W giv a lowr bound for th partial drivativ of th phas function in th x d variabl W hav x d ξ + v,c + ξx d m v, c ξ ξm x d Rcall that = m r, and ξ < c r 1 m ɛ/, hnc x d ξ + v,c + ξx d m v, c c r 1 m ɛ x d By intgrating by parts multipl tims, w dduc that I m, c 1 k c r 1 m k Thrfor, w concludd th claimd uppr bound 66 for th typ I vctors In ordr to prov th lmma it rmains to show C m m d S m, ci m, c 1, =1 c 1,

22 ASER T SARDARI whr th sum is ovr th ordinary typ I vctors c W not that th following sum ovr c Z d is boundd 1 1, c d+1 c 0 and as a rsult ovr th typ I intgral vctors Hnc, by choosing k larg nough in 66, w dduc our Lmma 6 Uppr bound on I m, c for th xcptional vctors c In this sction w assum that c is an xcptional vctor, and giv an uppr bound on I m, c Lmma 6 W hav I m, c [ m d 1 c d 1 ] min mr 1, 1, whr th implid constant in only dpnds on Ω and th fixd functions ψ 1 and ψ Proof W apply formula 64 and 69 and obtain I m, c = m v, d x T Av h, ywx d 1 i=1 c iũ i + β c, v m W us th wight function w dfind in 54, and obtain 611 I m, c = m By idntity 55, w hav h, yψ 1y ψ ũ d 1 i=1 c iũ i + β c, v m dũ1 dũ d 1 dy dũ1 dũ d 1 dy 1 + mβ = + m y d 1 i=1 m ũ i µ i By changing th variabls to x i := ũ i / for 1 i d 1 and using = m r, w obtain d 1 mβ = 1 + r y r x i µ i 1 By Lmma 5, if w 0 thn x i, y < C for som fixd constant C Hnc, by writing th Taylor sris of th suar root function, w hav 61 β = 1 [ d 1 ] m 1 r y m 1 r x i µ i + φx 1,, x d 1, y, whr 613 i=1 i=1 k φ = O k r 4 m 1, x i1 x ik for vry k 0 and any point insid th support of w W chang th variabls to x 1,, x d 1, y in formula 611, and rplac β with formula 61, I m, c m = d 1 h c, v y, yψ 1y mr 1 d 1 i=1 ψ x 1,, x d 1 c [ ix i m 1 r d 1 i=1 x i µ i] c, v + φ c, v dx 1 dx d 1 dy m

23 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 3 W dfin ξx 1,, x d 1, y : = Ly : = By Fubini s thorm, I m, c m I m, c m d 1 i=1 c [ ix i m 1 r d 1 m = d 1 d 1 i=1 x i µ i] c, v + φ c, v, ψ x 1,, x d 1 ξx 1,, x d 1, y h c, v y, yψ 1y mr 1 Lydy sup ψ1 y Ly h y [ C,C], y dy dx 1 dx d 1 By lmma 3 for l = k = 0, w dduc that h, y dy < C 1, whr C 1 is a constant indpndnt of / Rcall that ψ 1 Cc R Thrfor, 614 I m, c m d 1 sup Ly y [ C,C] W giv an uppr bound on th oscillatory intgral Ly W hav 615 ξx 1,, x d 1, y = 1 c i µ i c, v x i x i m mr + c, v φ x i Without loss of gnrality, w assum that r < ɛ 0, whr ɛ 0 dpnds only on th compact st Ω Lt c l = max 1 i d 1 c i and assum that 616 µ l c, v C mr c l 4, whr th constant C is dfind in Lmma 5 By 613, support of w Hnc, by choosing ɛ 0 small nough 617 φ c, v c l x l 4, φ x l = O r 4 m 1 in th By applying th inualitis 616 and 617 on uation 615, w obtain x l ξ c l m By inualitis 613 and 616 for vry k 0, w hav k ξ c l x k k l m By intgration by parts multipl tims in th x l variabl, w dduc that Ly A sup j ξ cl 0 j A+1 x j m l Finally, by 614 and 616, w dduc that I m, c m d 1 min A+1 A min [ c mr 1 [ cl m A, 1 ] A, 1 ] This concluds th lmma by assuming 616 It rmains to prov our lmma whn c l 4 < µ l c, v C, mr

24 4 ASER T SARDARI for vry 1 l d 1 Sinc a = v/ Ω and = mr 1 thn c l r c, a Lt a = d i=1 a i i thn by choosing ɛ 0 small nough, w dduc that 1/ c, a c d a d < c, a, and c d = max c i i Hnc, 618 c c, a Thrfor, x i x j ξx 1,, x d 1, y = 1 µ i c, v δi, j m mr + φ c, v, x i x j whr δi, j = 1 if i = j and δi, j = 0 othrwis By applying 613, w obtain ξx 1,, x d 1, y = δi, j µ i c, v c, v x i x j m r + O m r 4 W substitut a = v and mr 1 =, and gt ξx 1,, x d 1, y = δi, j µ i c, a c, a x i x j mr 1 + O mr 3 Finally by inuality 618 and th stationary phas thorm on th oscillatory intgral Ly, w obtain This concluds th lmma [ Ly d 1 c d 1 ] min mr 1, 1 63 Bounding th contribution of th xcptional vctors In this sction w bound th contribution of th xcptional vctors to th right hand sid of 310 Proposition 63 W hav m d S m, ci m, c c 0 =1 m 5 r 1 m ɛ r 1 m 1/ if d = 4, m d+1 r 1 m ɛ r 1 d 3 m if d 5 for any ɛ > 0, whr th sum is ovr th xcptional vctors c, and th implid constant in only dpnds on ɛ, Ω and th fixd functions ψ 1 and ψ W giv th proof of th abov proposition at th nd of this sction W writ whr m d S m, ci m, c = H 1 + H, =1 c 0 H 1 := c 0 c r 1 m m d S m, ci m, c, =1

25 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 5 and H := c 0 = c r 1 m m d S m, ci m, c By Lmma 4, S m, c = 0, unlss c αaλ for som scalar α Without loss of gnrality w assum that c αaλ mod m First, w bound th numbr of xcptional vctors satisfying this congrunc condition Lmma 64 Th numbr of xcptional intgral vctors c such that c αaλ mod m, for som α Z/mZ and c T, is boundd by T r 1 m ɛ Proof Fix α mod m with on of th m choics mod m Th proof is basd on th covring of R d by boxs of volum m d around th intgral points c whr c αaλ mod m By th dfinition of th xcptional vctors, c i < mmr 1 ɛ Thrfor, all th xcptional vctors with c < T li insid a box of volum T m d 1 mr 1 d 1ɛ By a covring argumnt, th numbr of xcptional vctors is lss than m T md 1 mr 1 d 1ɛ m d = T mr 1 d 1ɛ By choosing ɛ small nough in th dfinition of th xcptional vctors, w conclud th lmma Lmma 65 W hav 619 H 1 md+1 d 3 r 1 r 1 m ɛ[ r 1 m d ] m for any ɛ > 0, whr th implid constant in only dpnds on ɛ, Ω and th fixd functions ψ 1 and ψ Proof By Lamma 6, H 1 md+1 c m d S m, c r 1 m c By Proposition 41, w hav 60 m d d+1 Sm, c m ɛ r 1 m c 1+ɛ d 1 By Lmma 64, w dduc that c d 3 r 1 m ɛ[ r 1 m d ] c <r 1 m 1+ɛ Thrfor, H 1 md+1 d 3 r 1 r 1 m ɛ[ r 1 m d ] m By choosing ɛ small nough in th dfinition of th xcptional vctors c, w concluds th lmma Rmark 66 ot that if d 5, thn r 1 m d 5 = O1 and th uppr bound 619 is sharp Howvr, if d = 4 by applying inuality 60 in th proof, w los all

26 6 ASER T SARDARI th potntial cancllation in th variabl In fact, if w assum th suar root cancllation in th variabl, thn 61 H 1 md+1 d 3 r 1 r 1 m ɛ m for any d 4 Lmma 67 W hav 6 H md+1 d 3 r 1 r 1 m ɛ[ r 1 m d ] m for any ɛ > 0, whr th implid constant in only dpnds on ɛ, Ω and th fixd functions ψ 1 and ψ Proof By using th trivial bound I m, c md+1 and Proposition 41, w hav H md+1 m d S m, c 63 By Lmma 64, w obtain Hnc, c <r 1 m 1+ɛ c md+1 m ɛ r 1 m c d 3 +ɛ d 3 r 1 m ɛ[ r 1 m d ] 64 H md+1 d 3 r 1 r 1 m ɛ[ r 1 m d ] m This concluds th lmma Rmark 68 Similarly, if d 5 thn th uppr bound 6 is sharp, and if d = 4 by applying inuality 63 in th proof w los a suar root canclation in th variabl In fact, if w assum th suar root cancllation in th variabl, thn H md+1 d 3 r 1 r 1 m ɛ m for any d 4 Th abov inuality and inuality 61 implis Conjctur 13 Proof of Proposition 63 By Lmma 65, it follows that m d+1 r 1 m ɛ r 1 m if d = 4, 1/ H 1 r 1 m ɛ r 1 d 3 m if d 5 m d+1 Similarly, by Lmma 67, w hav H m d+1 r 1 m ɛ r 1 m if d = 4, 1/ m d+1 r 1 m ɛ r 1 d 3 m if d 5 Th proposition follows from th abov inualitis

27 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 7 7 Th main thorm Rcall th notations usd whil formulating Thorm 16, and th dfinition of w, λ in 310 In this sction w giv an asymptotic formula for w, λ, which implis Thorm 16 Th main trm of this formula coms from c = 0 First, w stimat th intgral I m, 0 Lt x 1,, x d b th coordinat systm dfind in Lmma 5 Dfin σ F, w = lim ɛ 0 ɛ/<yx<ɛ/ wx a, d dx 1 dx d ɛ By Lmma 5, it follows that thr xists constants c 1 and c, that only dpnd on Ω and th fixd smooth functions ψ 1 and ψ, such that 71 c 1 σ F, w c Lmma 71 Suppos that 1 1 ɛ Thn for any k > 0 I m, 0 = σ F, w md+1 + O k Proof Rcall th formula 37, and plug in c = 0 in that to obtain F mt + λ I m, 0 = h, m wmt + λdt 1 dt d Rcall th dfinition of w from 54 Chang th variabls from t 1, t d to x 1,, x d 1, y that wr introducd in Lmma 5, and apply 65 to gt m d+1 I m, 0 = h, yψ 1y ψ x 1,, x d 1 dx 1 dx d 1 dy By taking th intgration ovr y, and using Lmma 33, w hav I m, 0 = md+1 ψ x 1,, x d 1 ψ 1 0 dx 1 dx d 1 + O k By 54, for y = 0 By 55, y 1 x d = I m, 0 = md+1 = md+1 v, ψ x 1,, x d 1 ψ 1 0 d = x T Av wx yx=0 lim ɛ 0 W substitut ths valus and obtain x T Av y x d 1wx a, d dx 1 dx d 1 + O k ɛ/<yx<ɛ/ wx a, d dx 1 dx d ɛ = σ F, w md+1 + O k + O k This concluds th proof of our lmma In th following lmma, w show that th contribution of 1 ɛ ngligibl in 310 is

28 8 ASER T SARDARI Lmma 7 W hav m d S m, 0I m, 0 md+1 ɛ = 1 ɛ 3 d +ɛ Proof W hav I m, 0 md+1 By Proposition 41, w hav Hnc, w conclud th Lmma = 1 ɛ m d S m, 0 3 d +ɛ W invok [HB96, Pag 50; Lmma 31] Lmma 73 Hath-Brown W hav m d S m, 0 = S Ba,r + OT 3 d/+ɛ ɛ 1 T for any ɛ > 0 As a rsult 1 1 ɛ m d S m, 0 = S Ba,r + O 3 d/+ɛ for any ɛ > 0, whr th implid constant in O ɛ only dpnds on ɛ and th uadratic from F Proof of Thorm 16 Lt w b th smooth wight function dfind in 54 By Lmma 5, suppw B a, r Rcall 310 w, λ = C m m d S m, ci m, c =1 c From Proposition 63 and Lmma 7, w can drop th nonzro intgral vctors c and rstrict th sum ovr 1 1 ɛ, hnc for d 5 w, λ = C 1 ɛ m m d S m, 0I m, 0+O ɛ Ba,r d 1 d Ba,r d 3 ɛ, =1 and for d = 4, w hav w, λ = C 1 ɛ m m d S m, 0I m, 0 + O ɛ Ba,r 3 B a,r 3 1 ɛ =1 By Lmma 71, w hav I m, 0 = σ F, w md+1 + O k k for any k > 0 Thrfor, C 1 ɛ m =1 m d S m, 0I m, 0 = σ F, w d 1 Finally, from Lmma 73, if d 5 it follows that w, λ = σ F, ws Ba,r B a,r d 1 d 1 ɛ m d S m, 0 + O k k =1 1 + O ɛ Ba,r d 3 ɛ,

29 OPTIMAL STROG APPROXIMATIO FOR UADRATIC FORMS 9 and for d = 4, w hav w, λ = σ F, ws Ba,r B a,r O ɛ Ba,r 3 1 ɛ Sinc w is boundd and supportd insid B a, r, thn V Z B w, λ This concluds Thorm 16 Rmark 74 Sinc w drivd a smooth counting formula for th numbr of intgral points with a powr saving rror trm By a standard mthod in analysis majorants and minorants for th indicator function of a ball w can find two smooth wight function w 1 x and w x such that thy approximat from blow and abov th indicator function χx of th opn ball B a, r, i w 1 x χx w x In this way it would b possibl to giv a counting formula for Thorm 16 with a powr saving rror trm instad of th statd lowr bound Rfrncs [AC37] F C Auluck and S Chowla Th rprsntation of a larg numbr as a sum of almost ual suars, Proc Indian Acad Sci, 6:818, 1937 [BR1] Jan Bourgain and Zév Rudnick Rstriction of toral ignfunctions to hyprsurfacs and nodal sts Gom Funct Anal, 4: , 01 [Chi95] Patrick Chiu Covring with Hck points J umbr Thory, 531:5 44, 1995 [Da10] Dirk Damn Localizd solutions in Waring s problm: th lowr bound Acta Arith, 14:19 143, 010 [DFI93] W Duk, J Fridlandr, and H Iwanic Bounds for automorphic L-functions Invnt Math, 111:1 8, 1993 [Eic54] Martin Eichlr uatrnär uadratisch Formn und di Rimannsch Vrmutung für di Kongrunzztafunktion Arch Math, 5: , 1954 [GG13] Anish Ghosh, Alxandr Gorodnik, and Amos vo Diophantin approximation and automorphic spctrum Int Math Rs ot IMR, 1: , 013 [GG15] Anish Ghosh, Alxandr Gorodnik, and Amos vo Diophantin approximation xponnts on homognous varitis In Rcnt trnds in rgodic thory and dynamical systms, volum 631 of Contmp Math, pags Amr Math Soc, Providnc, RI, 015 [GG16] Anish Ghosh, Alxandr Gorodnik, and Amos vo Bst possibl rats of distribution of dns lattic orbits in homognous spacs J rin angw Math, 016 [Har90] Glyn Harman Approximation of ral matrics by intgral matrics J umbr Thory, 341:63 81, 1990 [HB96] D R Hath-Brown A nw form of th circl mthod, and its application to uadratic forms J Rin Angw Math, 481:149 06, 1996 [Hoo78] Christophr Hooly On th gratst prim factor of a cubic polynomial J Rin Angw Math, 303/304:1 50, 1978 [Kim03] Hnry H Kim Functoriality for th xtrior suar of GL 4 and th symmtric fourth of GL J Amr Math Soc, 161: , 003 With appndix 1 by Dinakar Ramakrishnan and appndix by Kim and Ptr Sarnak [Klo7] H D Kloostrman On th rprsntation of numbrs in th form ax + by + cz + dt Acta Math, 493-4: , 197 [LPS88] A Lubotzky, R Phillips, and P Sarnak Ramanujan graphs Combinatorica, 83:61 77, 1988 [Mal6] A V Malyšv On th rprsntation of intgrs by positiv uadratic forms Trudy Mat Inst Stklov, 65:1, 196 [Mar88] G A Margulis Explicit group-thortic constructions of combinatorial schms and thir applications in th construction of xpandrs and concntrators Problmy Prdachi Informatsii, 41:51 60, 1988 [RS17] Igor Rivin and asr T Sardari uantum chaos on random cayly graphs of Exprimntal Mathmatics, 00:1 14, 017 [Sar90] Ptr Sarnak Som applications of modular forms, volum 99 of Cambridg Tracts in Mathmatics Cambridg Univrsity Prss, Cambridg, 1990