Journal of Number Theory

Transcription

1 Journal of Numbr Thory Contnts lists availabl at ScincDirct Journal of Numbr Thory wwwlsvircom/locat/jnt Th sum of digits of squars in Z[i] Johanns F Morgnbssr Institut für Diskrt Mathmatik und Gomtri, Tchnisch Univrsität Win, Widnr Hauptstraß 8-0/04, A-040 Win, Austria articl info abstract Articl history: Rcivd 8 Octobr 009 Rvisd 3 Fbruary 00 Communicatd by K Soundararajan Kywords: Sum-of-digits function Gaussian intgrs Exponntial sums Følnr squncs Squars W considr th complx sum-of-digits function s q for squars with rspct to spcial bass q of a canonical numbr systm in th Gaussian intgrs Z[i] In particular, w show that th squnc αs q Z[i] is uniformly distributd modulo if and only if α is irrational Furthrmor w introduc spcial sts of Gaussian intgrs rlatd to Følnr squncs for which w can dtrmin th ordr of magnitud of th numbr of intgrs for which s q lis in a fixd rsidu class mod m This xtnds a rcnt rsult of Mauduit and Rivat to Z[i] Walsoimprovanstimat of Gittnbrgr and Thuswaldnr in ordr to show a local limit thorm for th sum-of-digits function of squars W can provid asymptotic xpansions for #{ Z[i] D N : s q = k} whr D N N N is a squnc of convx sts 00 Elsvir Inc All rights rsrvd Introduction and main rsults Throughout th papr w us th notation x for th xponntial function π ix IfA and B ar two sts, thn A B dnots th symmtric diffrnc of ths sts If q = a ± i choos a sign for a positiv intgr a, thnq = q = a + and N ={0,,, q } WdnotbyR m Z[i] a complt rsidu systm modulo m and writ x for th distanc from x R to its narst intgr Th symbol f g mans that f = Og and if not othrwis statd, th implid constant in th big O trm dpnds at most on q Th main motivation of this papr is to xtnd a rcnt rsult of Mauduit and Rivat [6] on th sum-of-digits function s q, whr q dnots an intgr Thy answrd an opn qustion posd by Glfond [7] in a papr of 968 concrning th uniform distribution modulo of th squnc αs q n n N, whr α R Furthrmor, thy studid th distribution of s q n in arithmtic progrs- addrss: johannsmorgnbssr@tuwinacat Th author is supportd by th Austrian Scinc Foundation FWF, grant S9604, that is part of th National Rsarch Ntwork Analytic Combinatorics and Probabilistic Numbr Thory 00-34X/$ s front mattr 00 Elsvir Inc All rights rsrvd doi:006/jjnt0000

2 434 JF Morgnbssr / Journal of Numbr Thory sions On can now ask whthr or not ths and similar rsults ar valid in th cas of Gaussian intgrs If q = a ± i, whr a Z +,thnvry Z[i] has a uniqu finit rprsntation of th form = j 0 ε j q j, ε j N, with ε j = 0for j gratr than or qual to som constant j 0 Th numbr ε j is calld th j-th digit of th numbr in th bas-q rprsntation systm s [4] Th sum-of-digits function s q is thn dfind by ε j 0 j Drmota, Rivat and Stoll [5] considrd Gaussian prims laying in a disc with radius N Thy showd th following thorm Thorm A Lt q = a ± ibapriminz[i], whra 8 is a positiv intgr and b, g Z, g Morovr, st d = g, a + a + and δ = d, ia +, whr th choic of th sign dpnds on th sign for q = a ± i Thn thr xists σ q,g > 0 such that # { p Z[i]: pprim, p N, s q p b mod g } = d g π in; b, d/δ + O q,g N σ q,g, whr π i N; b, d/δ dnots th numbr of Gaussian prims with p b mod d/δ and p N Concrning squars in th Gaussian intgrs, w stat a thorm of Gittnbrgr and Thuswaldnr [8], which dals with th asymptotic normality of th sum-of-digits function W st μ Q = Q / and σ Q = Q / Thorm B W hav, as N, { #{: < N} # < N: s q μ Q log Q N } < y Φy, σ Q log Q N whr Φ is th normal distribution function and runs through th Gaussian intgrs Local rsults of th sum-of-digits function and, mor gnrally, of block additiv functions on th Gaussian intgrs wr tratd by Drmota, Grabnr and Liardt [3] Thy provd that if k μ Q log Q N C log Q N for som C > 0 on has { # < N: s q = } π N k = πσ Q log Q N k μ Q log Q N σ Q log Q N + O log N In thir papr thy also usd an approach basd on rgodic Z[i]-actions and skw products to xtnd distributional rsults with rspct to so-calld Følnr squncs B n n N Th crucial proprty of such a squnc is that for all g Z[i] on has #B n g + B n = o#b n for a complt dfinition s [3, Sction 5] Thy showd for xampl that { lim # B n : s q + y } m mod M = n #B n M, with rspct to any Følnr squnc B n n N In what follows, w dfin κ-z[i] squncs which ar spcial Følnr squncs whr th rat of convrgnc coms into play

3 JF Morgnbssr / Journal of Numbr Thory Dfinition Lt κ baralnumbrsatisfying0< κ / A squnc D N N N of substs of Z[i] is calld a κ-z[i] squnc, if for all N N, i D N D N+, ii D N { Z[i]: max R, I N}, iii thr xists a constant c > 0, such that cn #D N, and iv #{D N r + D N } r N κ for all r Z[i] Th simplst xampls of /-Z[i] squncs ar Gaussian intgrs laying in squars with sid lngth N or discs with radius N It turns out that vry squnc of convx sts satisfying conditions i, ii and iii is a /-Z[i] squnc To b mor prcis, lt C N N N b a squnc of convx substs of C with C N C N+, C N { C: max R, I N} and such that th volum is gratr than cn for a positiv constant c > 0 Thn w hav that th squnc D N N N dfind by D N := C N Z[i] is a /-Z[i] squnc In this work w show distributional rsults for th sum-of-digits function of squars with rspct to κ-z[i] squncs Our main objctiv is to obtain information on th xponntial sum D N αsq In what follows w hav to assum that q = a ± i has not too small prim divisors, that is, vry divisor p q has to satisfy p 689 This rstriction coms into play whn considring th Fourir transform of th sum-of-digits function In contrast to th ral cas, thr aris som tchnical problms which maks it impossibl to covr gnral q S th introduction of Sction 43 for a mor prcis analysis of this problm From th proof of a rsult of Iwanic [] concrning th rprsntation of a + as almost-prims, it turns out that infinitly many Gaussian intgrs q = a ± i satisfy this condition W st A := {a N: ifp q = a ± i,thn p 689} ={36, 40, 54, 56, 66, 74, 84, 90, 94,} Thorm Lt a A and α RFurthrmorltκ b a ral numbr satisfying 0 < κ / and D N N N a κ-z[i] squnc Thn thr xists a constant c q,κ > 0,suchthat αsq q log N ωq/+ N c q,κ a +a+α, D N whr ωq dnots th numbr of distinct prim divisors of q W can driv from this thorm th ordr of magnitud of th numbr of squars whos sumof-digits function valuation lis in a fixd rsidu class modulo som intgr m Furthrmor w gt that th squnc αs q is uniformly distributd modulo if and only if α R \ Q W st Q b, s = #{ R s : b mod s} Thorm Lt a A and b, g Z, g Std= g, q FurthrmorltD N N N b a κ-z[i] squnc with 0 < κ / Thn thr xists a constant σ q,g,κ > 0 such that # { D N : s q b mod g } = #D N g Q b, d + O q,g N σ q,g,κ

4 436 JF Morgnbssr / Journal of Numbr Thory Thorm 3 Lt a A Thn th squnc αs q Z[i] is uniformly distributd modulo if and only if α is irrational Th nxt thorm provids asymptotic xpansions for #{ D N : s q = k} whnvr D N is a /-Z[i] squnc If D N is a disc with radius N, it can b sn as a local vrsion of Thorm B Thorm 4 Lt a A and D N N N b a /-Z[i] squnc Thn w hav uniformly for all intgrs k 0, #D N # { D N : s q = k } = Q k, q πσ Q log Q N log log N k / + O log N, whr k = k μ Q log Q N σ Q log Q N Not that this last thorm is trivial in th cas that th numbrs k and μ Q log Q N diffr too much If w considr th disc with radius N and count th numbr of squars whos sum-of-digits function s q quals th xpctd valu μ Q log Q 4, w gt th following corollary, which is a dirct consqunc of Thorm 4 Th proof is straightforward and w omit it sinc it is vry similar to th proof of Thorm in [4] Corollary Lt a A W hav, as N, { # < N: s q = μ Q log Q 4 } π N μq log Q N log log N = R log Q N a + O, + a + log N whr Rt dnots a positiv priodic function with priod Plan of th papr Th papr is organid as follows In th nxt sction w trat Gauss sums in Z[i] and th discrt Fourir transform of αs q In Sction 4 w prov th main rsult of this work, i, w driv an stimat of th xponntial sum αsq 3 D N Th undrlying mthod which w us is basd on th work of Mauduit and Rivat [6] on th ral sum-of-digits function of squars In a first stp Sction 4 w transform th problm in such a way that w ar abl to work with diffrncs of th form αs q + r αs q 4 In ordr to do so, w prov a van dr Corput typ inquality which is th ky lmma for working with κ-z[i] squncs It turns out that this lmma is also of grat importanc for th proof of Thorm 4 Th advantag of xprssions of th form 4 is that th highr placd digits of do not contribut to th xact valu in th most cass With th hlp of th addition automaton, w show that w can rplac th sum-of-digits function with a truncatd vrsion of this function carry lmma In contrast to th ral cas and to th prim numbr cas in Z[i] this taks up a larg part of th

5 JF Morgnbssr / Journal of Numbr Thory proof of Thorm In Sction 4 w valuat sums containing linar xponntial sums Ths rsults nabl us to find an uppr bound of 3 which only dpnds on th discrt Fourir transform of th truncatd sum-of-digits function Sction 43 finally contains th last stps of th proof of Thorm In Sction 5 w first stat and prov som basic proprtis of congruncs in Z[i] on th on hand and κ-z[i] squncs on th othr hand With hlp of ths rsults and Thorm w giv for th sak of compltnss straightforward proofs of Thorm and Thorm 3 In th last sction w prov Thorm 4 In ordr to driv asymptotic xpansions for #{n D N : s q n = k}, w procd as in [4], whr Drmota, Mauduit and Rivat showd a local rsult for prims in Z Although th calculations ar much mor involvd, w can improv th basic mthod givn in [4, Sction 4] to obtain a bttr rror trm Th starting point of this mthod is th rlation { # D N : s q = } k = 0 Sα αk dα, whr Sα dnots th sum occurring in 3 W split th intgral up into two diffrnt domains If α is not too clos to l/a + a +, l = 0,,a + a +, w us Thorm to obtain a ngligibl rror trm Th rmaining part of th intgral yilds th main trm in Thorm 4 To calculat this part, w us probabilistic mthods in ordr to succd In Sction 6 w show that th sum-ofdigits function can b wll approximatd by a sum of indpndnt and uniformly distributd random variabls W us an ida of Bassily and Kátai [] and improv som rsults of Gittnbrgr and Thuswaldnr [8], which dal with asymptotic normality in th Gaussian intgrs Again, xponntial sums play an important rol In Sction 6 w finally finish th proof of Thorm 4 3 Auxiliary rsults 3 Gauss sums in Z[i] Gauss introducd th sum m n=0 bn /m, which is nowadays calld a quadratic Gauss sum It is wll undrstood and a lot of diffrnt gnraliations hav bn studid s for xampl [,], or [3, Chaptr 34] On possibl gnraliation which w will nd latr on, ar sums of th form Gb,l;m = bn tr + ln, m n R m whr b, l, m Z[i] with m 0 and R m is a complt rsidu systm modulo m tr = + = R dnots th trac of C Lmma Lt a A,q= a ± iandltb,l Z[i] with b, q = Thnwhavforr, G b,l; q r = q r Rmark Basd on a thorough analysis of th corrsponding proof for Gauss sums in Z s [, Chaptr 74], on can show th following rsult: If b,l,m Z[i] ar such that m 0 and b,m =, thn on has Gb,l;m m Proof of Lmma W bgin with th following obsrvation: If m,m =, thn w hav Gb,l;m m = Gbm,l;m Gbm,l;m 5

6 438 JF Morgnbssr / Journal of Numbr Thory This is a consqunc of th fact that if j and k run through a complt rsidu systm modulo m, rsp m, thn th numbrs jm + km run through a complt rsidu systm modulo m m In what follows w will show that for vry prim divisor p of q w hav G b,l; p r = p r 6 Combind with 5, this provs th dsird rsult First w show that Gb,l; p r = Gb, 0; p r Sinc a A and p q = a ± i, whav + i, p = Thus, 4b, p r = and 4b has an invrs lmnt modulo p r,say b Rplacingn by n + bl in th indx of summation yilds G b,l; p r = = n R p r tr bn + ln p r bl tr p r n R p r = tr n R p r bn p r tr bn bl + ln bl p r Nxt not, that if w st Rp r, Ip r = d, thnd has to b of th form p δ for som δ 0 But sinc p q, this can only hold tru for δ = 0 Thus, th st {n Z: 0 n < p r } forms a complt rsidu systm modulo p r and w can writ G b, 0; p r = 0 n< p r = 0 n< p r bn tr p r cn p r, = 0 n< p r tr bn p r whr c =R p r b Sinc p is a prim divisor of q w hav that p is a prim in Z and satisfis p > Nxt w show that c and p ar coprim St d =I p r b If c, p, thn c, p = p This implis that p c, p c and p d, which in turn implis that p d and thus p c + id It follows that p p, which is impossibl sinc p > This allows us to us wll-known rsults for Gauss sums in Z Employing [, Lmmas 7, 73 and 75], w obtain G b, 0; p r = 0 n< p r cn p r = p r, p r which finally provs 6 and th statmnt of th lmma Lmma Lt a A and q = a ± i Furthrmor, lt b,l Z[i] and r Std= b, q r Thnwhav G b,l; q r = 0 if d l, and G b,l; q r = dq r if d l Proof Th proof is straightforward and is omittd hr For dtails s for xampl [6, Proposition and Proposition ]

7 JF Morgnbssr / Journal of Numbr Thory Fourir analysis of F λ Throughout what follows w dnot by F λ ={ λ j=0 ε jq j : ε j N } th finit non-scald approximation of th fundamntal rgion of th numbr systm, which is a complt systm of rsidus mod q λ with #F λ = Q λ Th discrt Fourir transform F λ, α of th function u αs q is dfind for all h Z[i] by F λ h,α = Q λ u F λ αs q u tr huq λ Sinc F λ, α is priodic with priod q λ and F 0 h, α =, w hav Fλ h,α = Q λ λ ϕ Q α tr hq j, 7 j= whr th function ϕ k t is dfind for all k by { sinπkt / sinπt, t R/Z, ϕ k t = k, othrwis Th following lmma collcts som basic facts of ϕ k t Th proof can b found in [6, Lmmas 3, 5] and [5, Lmmas 4, 5] Lmma 3 Lt k b an intgr Thn, th following claims hold tru: i Th function ϕ k t is priodic of priod, monotonically dcrasing on [0, /k] andwhavforδ [0, /3k], max ϕ kt ϕ k δ k t δ ii If t 6 π k,thnwhav iii Lt η k b dfind by k η k = max t R k ϕ k t k xp k π t 6 0 r<k ϕ kt + r k Thn k η k k sin π k + π log k π In particular, w hav < η 689 < and η k < η 689 for k > 689 iv Lt k 3If3 R kandr k, thn max t R k 0 r<r ϕ k t + r R η R R If R = and R k, thn th right-hand sid of th givn stimat can b rplacd by 3/ < 3 < 34 < η 5

8 440 JF Morgnbssr / Journal of Numbr Thory Th nxt lmma is a variant of [5, Corollary 64] and givs an xplicit uniform uppr bound for F λ h, α Lmma 4 Lt c Q = π 7 log Q Q Q 4 Thnwhavforλ 3, Fλ h,α π /54Q Q c Q a +a+α λ Proof Sinc /3Q 6/π Q /, w obtain from point i and ii of Lmma 3 that w hav ϕ Q t Q xp Q π t if t /3Q and ϕ 6 Q t ϕ Q /3Q Q xp Q π /3Q t if t > /3Q W obtain that for all t R th following stimat 6 holds: ϕ Q t Q xp π Q 7Q t Carrying out xactly th sam stps as in th proof of [5, Corollary 64], w gt Fλ h,α xp λ Q π Q a + 7Q a + α π /54Q Q c Q a +a+α, which finishs th proof of this lmma Rmark With a thorough analysis of th proof of [6, Lmma 6] for Gaussian intgrs with thr adjacnt trms instad of two, on can improv th xplicit givn constant c Q to b c Q = π 8 log Q Q a + a + Th last lmma of this sction givs finally an L typ uppr bound of F λ Itisanimprovmnt of [5, Lmma 66] and nabls us to considr composit bass in Thorm instad of just prim bass s Sction 43 Lmma 5 Lt a > and q = a ± i Furthrmor, lt b Z[i], α R, 0 δ λ and k Z[i] with k q λ δ and q k Thn w hav h R q λ h b mod kq δ Fλ h,α k η 5 Q η 5λ δ Fδ b,α If a A,thnη 5 can b rplacd by η 689 Proof Th proof follows th lins of th proof of [5, Lmma 7] W just giv a rough outlin of thos parts which ar ssntially th sam as in th ral cas and trat th stps which ar crucial in th stting of Gaussian intgrs in dtail If δ = λ, thn th condition k q λ δ implis k = and th statmnt holds trivially If δ<λ, thn w dfin d θ = q θ, kq δ whr w choos on gratst common divisor and u θ = q θ /d θ whnvr δ θ λ Ifwstρ θ = d θ /d θ, thn on can show as in th ral cas that th following claim holds tru: ρ θ is a Gaussian intgr satisfying ρ θ q and q, ρ θ q Our main goal is to show that for δ<θ λ,

9 JF Morgnbssr / Journal of Numbr Thory Fθ h,α ρ θ η 5 Q η 5 Fθ h,α, 8 h R q θ h b mod d θ h R q θ h b mod d θ whr η 5 can b rplacd by η 689 if vry prim divisor p of q = a ± i satisfis p 689 Th statmnt of th lmma follows thn th sam way as in th proof of [5, Lmma 7] In ordr to show th rsult, w start with rwriting th lft-hand sid of 8 W hav Fθ h,α = Fθ b + ud θ,α h R q θ h b mod d θ = u R u θ Fθ b + vd θ,α v R qu θ v 0 modρ θ = Fθ b + u + wuθ d θ,α u R u θ w R q u+wu θ 0modρ θ Sinc u θ d θ = q θ and F θ, α is priodic of priod q θ, w obtain by th product rprsntation of F θ s 7 that Fθ h,α = Fθ b + ud θ,α Eθ, 9 h R q θ h b mod d θ u R u θ whr Eθ is dfind by Eθ = Q ϕ Q w R q u+wu θ 0modρ θ α tr b + udθ q θ w tr q Nxt w sk for an uppr bound of Eθ, which is th main part of this proof Sinc d θ ρ θ, u θ = d θ, q θ = q θ, kq δ, q θ = d θ, w s that ρ θ, u θ = This implis that u θ has an invrs modulo ρ θ say ũ θ and w can rwrit th condition u + wu θ 0 modρ θ to w uũ θ mod ρ θ Sinc w hav to considr Gaussian intgrs w lying in a complt rsidu systm modulo q, w can choos w = uũ θ rρ θ, r R q/ρθ Wobtain Eθ = Q = Q r R q/ρθ ϕ Q r R q/ρθ ϕ Q α tr b + udθ q θ uũθ tr rρ θ q α b + udθ tr q θ + uũ θ + r q tr q/ρ θ St q = q/ρ θ and d = R q, I q Thn d has to b a divisor of q and thus of q Butsincq = a ± i, this is only possibl if d = Hnc, {0 r < q } forms a complt rsidu systm modulo q Wgt Eθ = ϕ Q α b + udθ Q tr q θ + uũ θ + r R q q q 0 r< q

10 44 JF Morgnbssr / Journal of Numbr Thory Not that R q, q = R q, R q +I q = R q, I q R q, I q = R q, I q = Thus, w hav R q, q = and it follows that Eθ = Q 0 r< q ϕ Q α b + udθ tr q θ + uũ θ + r q q Using point iv of Lmma 3 w obtain Eθ ρ θ η q Q η q Sinc q = q/ρ θ is a divisor of q = a ± i w hav q = or q 5 In both cass point iv in combination with point iii of Lmma 3 implis Eθ ρ θ η 5 Q η 5 If a A, thn q 689 and w can rplac η 5 by η 689 Togthr with 9 this shows th dsird rsult 4 Proof of Thorm Rcall that w hav q = a ± i with a Z + and α R Wst f n = αs q n and dfin for vry ral positiv numbr B th st Ξ B Z[i] by Ξ B := { n Z[i]: max Rn, In B /} Lt ν Z + b dfind by Q ν < N Q ν Thn w will show that S := f n q ν ωq/+ Q c q,κ a +a+α ν, 0 n D N whr c q,κ is a positiv constant and ωq dnots th numbr of distinct prim divisors of q Of cours, this provs Thorm 4 Van dr Corput typ inquality and carry lmma W bgin with a lmma that contains a van dr Corput typ inquality for Gaussian intgrs It allows us to work with diffrncs of th form αs q + r αs q, which is important for th furthr stps of th proof Morovr, it has th rmarkabl proprty that w can nlarg th domain of summation without producing a too big rror trm On th lft-hand sid of inquality w sum up ovr Gaussian intgrs laying in D N, whil on th right-hand sid w sum up ovr all Gaussian intgrs laying in a squar which contains D N It prmits us to calculat spcial xponntial sums that appar in latr parts of th proof s Lmma 9 and Lmma 0 Th ida of th lmma and th proof of it is inspird by [6, Lmma 5]

11 JF Morgnbssr / Journal of Numbr Thory Lmma 6 Lt B b a ral numbr and N a positiv intgr satisfying N B Furthrmor lt n n Z[i] b complx numbrs with absolut valu and D M M N b a κ-z[i] squnc Thn w hav for any ral numbr R, n D N n N R r / n+r n R + + N κ R n,n+r Ξ B r R Proof Bfor w start th main part of th proof, w obsrv that #D N #Ξ N N Nxtwtak for convninc n = 0forn / Ξ B and put T R = #{0 r R} Sinc th absolut valus of th considrd complx numbrs ar, w obtain using th proprtis of th κ-z[i] squnc T R n n D N n+r r R n D N n n+r r R n D N n D N { # D N r + D N } r R r N κ T RRN κ r R Thus, w hav n D N n T R n+r + N κ R n D N r R Using th Cauchy Schwar inquality, w obtain n+r #D N n+r n D N r R N = N n D N r R 0 r, r R n Z[i] 0 r R wr n Z[i] n+r n+r n+r n, whr wr = #{r, r, 0 r, r R: r r = r} R + R + r SincT R R,w finally gt n D N n N R r n+r n R + n,n+r Ξ B r R / + N κ R Bfor w start to stimat th sum S s 0, w stat a rsult that provids information about th numbr of digits of a givn numbr Z[i] for a proof s for xampl [9, Proposition 6] Lmma 7 Lt l 0 b th smallst numbr satisfying = 0 j<l ε jq j with ε j N Thn thr xists a constant c which only dpnds on q,suchthat log Q c l log Q + c

12 444 JF Morgnbssr / Journal of Numbr Thory WmployLmma6withB = Q ν and R = q ρ /, whr ρ is an intgr satisfying ρ ν/3, and gt N S Q ρ r q ρ Q ν ρ + Q ν r q ρ + f n + r f n / + N κ q ρ n,n+r Ξ Q ν max r q ρ f n + r f n / + Q ν νκ ρ/ n,n+r Ξ Q ν In th last stp, w sparatd th cas r = 0 and r 0 Additionally, w gt an rror trm OQ 3ν+ρ/ insid th squar root whn rmoving th summation condition n + r Ξ Q ν But sinc w hav assumd ρ ν/3, this trm can b nglctd Hnc, w obtain S Q ν ρ/ + Q ν νκ ρ/ + Q ν/ max r q ρ f n + r f n / 3 n Ξ Q ν In a nxt stp, w want to us th fact that w ar now daling with xprssions of th form f n + r f n Ifr is small in comparison to n, th highr placd digits of n + r and n do not diffr in most of th cass In ordr to show this, w dfin a truncatd sum-of-digits function tims th constant α, namly, λ f λ = α ε j, whr ε j, j 0 ar th digits of in th bas-q rprsntation Th advantag of this function is that it is priodic with priod q λ, i, for any d, Z[i], whav j=0 f λ + dq λ = f λ For a proof of this statmnt, s [5, Proposition 4] Nxt, w brifly rcall that th addition in th Gaussian intgrs can b ralid by an automaton s [9] For our furthr xplanations w rstrict ourslvs to bas q = a + i, thcasq = a i is similar Th addition automaton in bas q = a + i is drawn in Fig Th digits of th sum ar associatd to a walk which finishs in on of th two accpting stats [ ] Starting at nod P, it prforms addition by and starting at nod R it prforms addition by a i Th labling j k mans that th automaton rads a digit j and has k as output Th nxt lmma givs an uppr bound of th numbr of cass, whr it maks a diffrnc if w us th normal or th truncatd sum of digits function Not, that th rasoning of th proof of th corrsponding carry lmma Lmma 4 in [5] dos not suffic for our purpos Nvrthlss, w also us th fact that th addition can b handld by an automaton Lmma 8 Lt r Z[i] with r < Q ρ WdnotbyEr, ν, ρ th st of Gaussian intgrs such that Ξ Q ν and f + r f f ν+ρ + r f ν+ρ 4

13 JF Morgnbssr / Journal of Numbr Thory Fig Addition automaton in bas q = a + i Thn w hav #Er,ν,ρ Q ν γρ, whr 0 < γ < is a constant only dpnding on q Proof By Lmma 7 w know that thr xists a constant c only dpnding on q such that for all Ξ Q ν th numbr has ν + c digits, r + r has ν + ρ + c digits and if w has l digits, thn w hav that q l c w q l+c W can assum that ρ > 4c th statmnt is trivial in th convrs cas Th main ida of th proof of this lmma is that vry Gaussian intgr Ξ Q ν can b uniquly writtn as = m 0 + q ν+ρ+c m + q ν+ρ m, whr m 0 F ν+ρ+c, m F ρ c and m F ν+c ρ Wst T = { t F ρ c : t = m for som Er,ν,ρ }, and N t = { Ξ Q ν : m = t and Er,ν,ρ } In what follows w show that #T Q γ ρ, 5 whr γ is a positiv constant satisfying γ < and

14 446 JF Morgnbssr / Journal of Numbr Thory #N t Q ν ρ 6 Th statmnt of th lmma is thn a dirct consqunc sinc #Er,ν,ρ = t T #N t t T Q ν ρ Q ν γ ρ Stting γ = γ > 0 provs indd th dsird rsult W procd in two stps First w show 5, i, w bound th numbr of possibl numbrs m which lad to a carry propagation Thrfor lt Er, ν, ρ W can writ + r + r = m 0 + r + r + q ν+ρ+c m + q ν+ρ m If w considr only th first part of th sum, w obtain that m 0 + r + r = m + q ν+ρ+c x + iy, whr m F ν+ρ+c is som Gaussian intgr dpnding on and r and x, y Z with x = O and y = O This is a consqunc of Lmma 7 Nxt w claim that f m + x + iy f m f ρ c m + x + iy f ρ c m 7 First not that f m = f ρ c m If w assum quality in 7, w conclud that m + x + iy has at most ρ c digits It follows that f + r f = f m + q ν+ρ+c x + iy + m + q ν+ρ m f m 0 + q ν+ρ+c m + q ν+ρ m = f m + q ν+ρ+c x + iy + m f m 0 + q ν+ρ+c m Th intgrs in th last xprssion hav at most ν + ρ digits and w obtain f + r f = f ν+ρ m + q ν+ρ+c x + iy + m f ν+ρ m0 + q ν+ρ+c m = f ν+ρ + r f ν+ρ This contradicts Er, ν, ρ and 7 holds tru indd From now on w assum that x + iy = y a i + x ay with y > 0 and x < ay all othr cass ar similar Using an ida dvlopd in [9, Proposition 4], th lft-hand sid of 7 can b writtn as f m + x + iy f m = f m + x + iy f m + a i + x + iy + f m + a i + x + iy f m + a i + x + iy + + f m + y a i + x + iy f m + y a i + x + iy

15 JF Morgnbssr / Journal of Numbr Thory Fig Calculation of S m + f m + y a i + x + iy f m + y a i + x + iy + + f m + y a i + x + iy + f m + y a i + x + iy f m + y a i + x + iy + x + ay f m Th numbr of diffrncs is O Sinc on can writ th right-hand sid of 7 in th sam way rplacing f by f ρ c, w dduc that 7 can b only fulfilld if on of th diffrncs is not qual to th corrsponding on with f rplacd by f ρ c Furthrmor ach of thm is of th form f x f x + u, whr u is ithr a i or Thus w ar intrstd in th numbr of cass whr th addition x x + u givs ris to a carry propagation Th sam rasoning as in [5, bottom of pag 39] shows cf [0, Proposition ] that this numbr is boundd by O ξ ρ, whr ξ =Q γ with γ < W gt that th numbr of possibl Gaussian intgrs t = m, such that thr occurs a carry propagation is boundd by OQ γ ρ and it finally follows that #T = OQ γ ρ For th scond stp proving th stimat 6 w fix som t F ρ c Wst S m := { : = m 0 + q ν+ρ+c t + q ν+ρ m, m 0 F ν+ρ+c } This allows us to writ N t m F ν+c ρ S m, and hnc #N t m F ν+c ρ #S m 8 If m = 0, w hav S 0 {: q ν+ρ }, and w obtain that #S 0 q ν+ρ Form 0, w can bound th cardinality of S m by #S m q ν+ρ ν+ρ t + q ρ c m + q + 9 t + q ρ c m To s this, considr Fig Th dashd domain in th first on contains th st of all possibl squars th solid circl has radius q ν+ρ+c and th chckrd domain contains th st S m On possibility to find an uppr bound of th cardinality of S m is to calculat th ara of th chckrd domain plus

16 448 JF Morgnbssr / Journal of Numbr Thory an additional ara if th Gaussian intgrs li on th bordr Th nxt pictur shows this rgion For radability, w dnot th angl by ϕ and writ a = q ν+ρ+c t + q ρ c m q c and b = q ν+ρ+c t + q ρ c m + q c First w calculat th ara of th circular ring sgmnt Th innr radius is a / and th outr radius is b + / W gt th ara b a + b + a ϕ b a + b ϕ 0 Th rmaining domain thicknss / has ara b a + b a + a Furthrmorwhavϕ t + q ρ c m not that t + q ρ c m q ρ c > q c Adding th xprssions 0 and togthr and using th stimats abov, w finally obtain 9 In ordr to avoid problms arising from th dnominators, w split th sum in 8 up into two parts Th first on contains all intgrs m satisfying m q ν 3ρ thy ar all in F ν ρ+c From 9 follows, that #S m q ν+ρ for all m 0 Using this crud uppr bound, w obtain 0< m q ν 3ρ S m 0< m q ν 3ρ q ν+ρ q ν ρ = Q ν ρ Now, w valuat th rmaining part in 8 Sinc m > q ν 3ρ, w hav that t +q ρ c m q ρ m Using 9, w gt m F ν ρ+c m > q ν 3ρ #S m m F ν ρ+c m > q ν 3ρ m F ν ρ+c m > q ν 3ρ ν+ρ q ν+ρ t + q ρ c m + q t + q ρ c m + q ν 0< m q ν 4ρ+c m + q ν/ m + / q ν m + q ν/ m + / Thus w hav to dal with sums of th form 0< m N m α, whr α {/, } A first crud stimation shows that 0< m N Calculating th sum of th right-hand sid, w gt N m α r α r=

17 Hnc w obtain 0< m q ν 4ρ+c q ν JF Morgnbssr / Journal of Numbr Thory < m N m + q ν/ m / + { N, if α =, m α N 3/, if α = / q ν q ν ρ + q ν/ q 3ν 6ρ + q ν 4ρ Q ν ρ This shows stimat 6 and th proof of Lmma 8 is finishd For furthr considrations, w st λ = ν + ρ Rplacing f by f λ givs a total rror of OQ ν γρ/ Thus, by 3 w hav S Q ν γρ/ + Q ν νκ ρ/ + Q ν/ max S r,ν,ρ /, 3 r q ρ whr S r,ν,ρ = n Ξ Q ν f λ n + r f λ n 4 Calculations yilding to th Fourir transform Nxt w stimat som spcial xponntial sums in ordr to obtain xprssions containing Gauss sums and th Fourir transform Th undrlying ida of th following stps coms from [6, Sction 54] but th stimats of xponntial sums turn out to b much mor involvd To bgin with, w stat for th sak of compltnss a wll-known rsult about linar xponntial sums s for xampl [8, Lmma ] Lmma 9 Lt q = a ± iwitha Z +, λ 0 and h Z[i]Sth/q λ = r + is, r, s RThnwhav h N N tr q λ min N, r, s, r s Ξ N Lmma 0 Lt m Z[i]\{0} and R m b a complt rsidu systm modulo m Morovr, lt N N Thnw hav h R m min N N, Rh/m, N Ih/m, Rh/m Ih/m 6N + 64 N m log m +64 m log m Rmark 3 This lmma is an improvmnt of [8, Lmma 6], whr Gittnbrgr and Thuswaldnr dal with similar sums Thy us th Koksma Hlawka inquality to obtain an rror trm of th form ON m + m log N, which suffics for th proof of thir main rsult In our cas, th trm N m is too big and w hav to us othr idas in ordr to succd

18 450 JF Morgnbssr / Journal of Numbr Thory Proof of Lmma 0 Sinc th summands ar priodic with priod m, it dos not mattr ovr which rsidu systm w sum Lt us dnot th considrd sum with T Not that R = Z[i] {α + iβm: / α,β < /} is a complt rsidu systm modulo m and st ˆR ={α,β [ /, / : α + iβm R} Thn, w can writ T = α,β ˆR N N min N, α, β, α β In a nxt stp, w tssllat th squar [ /, / with small squars of sid lngth / m Thrfor, w st and [ h ˆR h,k := ˆR m, h + m [ k m, k + m, ˆR h,k := ˆRh,k ˆR h,k ˆR h, k ˆRh, k W obtain T = m h=0 m k=0 N min N, α, α,β ˆR h,k N β, α β Not, that th innr sum has lss or qual 4 m / m + 6 summands W distinguish btwn thr diffrnt cass If h = k = 0, w us N as an uppr bound of th summands If h 0 and k = 0, w us N h/ m rsp N k/ m whn h = 0 and k 0 and h/ m k/ m in th rmaining cas Thus w hav T 4 m / m + N + N m k= k/ m + m k= k/ m 6N + 64 N m log m +64 m log m Lmma Lt q = a ± iwitha Z +, λ 0 and n n Z[i] b complx numbrs of priod q λ Thnw hav for < N Q λ, n ν max hn h F λ n tr q λ n Ξ N n F λ Proof Using th priodicity of th considrd complx numbrs, w can writ n = n Ξ N n n F λ Ξ N = Q λ h F λ Ξ N h F λ Q λ tr tr h q λ hn q λ hn n tr q λ n F λ

19 JF Morgnbssr / Journal of Numbr Thory Rcall that λ = ν + ρ and ρ ν/3 s and Thus w hav log q λ ν Applying th two prvious lmmas yilds th dsird rsult As indicatd abov, f λ is priodic with priod q λ Using th Fourir transform F λ h, α compar with Sction 3, w obtain S = h,h F λ F λ h,αf λ h,α Employing Lmma with N = Q ν yilds S ν n Ξ Q ν tr Fλ h,αf λ h,α h n + r + h n h,h F λ max h l F λ tr n + r + h n + ln q λ n F λ Using th notion of th quadratic Gauss sums, w can writ S ν q λ Fλ h,αf λ h,α max G h + h, rh + l; q λ l F λ h,h F λ Lmmagivs S ν Q λ/ max d l F λ d q λ h,h F λ h +h,q λ =d d rh +l Fλ h,αf λ h,α, whr w only sum ovr on of th four possibl associatd lmnts of a divisor d 43 Final stimats for th proof of Thorm For furthr calculations, w only considr th cas whr q = a ± i with a A Thisrsultsfrom th fact that w mploy Lmma 5, which is not sufficint if q has small prim divisors In particular, th constant η k has to b smallr than /4 svral considrations in th following stps of th proof nd this assumption First w rplac th summation condition h + h, k λ = d by th lss rstrictiv on h + h 0modd Th condition rh + l 0 mod d 4 can b rwrittn as follows: St d = r, d = r, d sinc d q λ,whavd, + i = Thn d l, sinc othrwis 4 cannot hold If w st r = r/ d and l = l/ d, thn 4 is quivalnt to r h + l 0 mod d/ d Th intgr r has an invrs lmnt modulo d/ d which w call r If w finally st l = r l,thnw can writ th last quation as h + l 0 mod d/ d

20 45 JF Morgnbssr / Journal of Numbr Thory W procd with sparating th sum according to th valu of ν q d, whr ν q d is th uniqu intgr δ, such that q δ d but q δ+ d St Thn w can writ ρ log Q ρ Q := 5 log 689 S ν Q λ/ max l F λ S + S 3, 6 whr S = 0 δ ρ Q k q λ δ q k kq δ h,h F λ h +h 0modkq δ h +l 0modkq δ /kq δ,r Fλ h,αf λ h,α, and S 3 = ρ Q <δ λ k q λ δ q k kq δ h,h F λ h +h 0modkq δ h +l 0modkq δ /kq δ,r Fλ h,αf λ h,α In ordr to find an uppr bound of S, w rplac th condition h + l 0modkq δ /kq δ, r by th wakr condition h + l 0 mod k/k, r This is allowd sinc k/k, r is a divisor of k k, r kqδ, rq δ kq δ = kqδ, r kq δ, r W can writ S 0 δ ρ Q k q λ δ q k kq δ h F λ h l mod k/k,r Fλ h,α h F λ h h mod kq δ Fλ h,α Sinc th Fourir transform is priodic with priod q λ in its first argumnt and sinc th moduli occurring in th summation conditions ar divisors of q λ, w can sum ovr an arbitrary complt congrunc systm modulo q λ Using Lmma 5 twic yilds to S 0 δ ρ Q = Q η 689λ k q λ δ q k kq δ Q η 689 λ k k, r 0 δ ρ Q Q / η 689δ k q λ δ q k η689 Q η 689λ δ k η 689 k 4η 689 k, r η 689

21 JF Morgnbssr / Journal of Numbr Thory Not, that k, r r q ρ Furthrmor,whavforvryk q λ δ with q k that k = k, q λ λ δ k, q λ δ q = Q λ δ log Q Now w nd th first tim that η 689 < /4 If w dnot by τ m th numbr of divisors of m, thn w obtain S Q η 689λ 0 δ ρ Q Q δ/ η 689 τ q λ δ Q = τ q λ Q λ 4η 689 log Q 689+ρη 689 λ δ log Q 689 4η 689 Q ρη δ ρ Q Q δη η 689 log Q 689 Sinc ρ ρ Q and log Q 689 4η 689 / 4η 689 / < 00000, w gt S τ q λ Q λ 4η 689 log Q 689+ρ Q 8 Nxt w stimat S 3 Sincδ>ρ Q, w hav that kq δ, r q ρ Q kq δ This follows from th fact that r q ρ and that vry prim divisor p of q satisfis ν p kq δ, r ρ log Q ρ log p Q cf [6, Sction 55] Hnc, w ar allowd to rplac th summation condition h + l 0modkq δ /kq δ, r by th lss rstrictiv condition h + l 0modkq δ ρ Q Whav S 3 ρ Q <δ λ k q λ δ q k kq δ h F λ h l mod kq δ ρ Q Fλ h,α h F λ h h mod kq δ Fλ h,α Using Lmma 5 in combination with Lmma 4, w gt Fλ h,α Fλ h,α h F λ h l mod kq δ ρ Q h F λ h h mod kq δ k 4η 689 Q λ δ+ρ Q η 689 δ ρ Q c Q a +a+α +λ δη 689 δc Q a +a+α, whr c Q is th positiv constant dfind in Lmma 4 It follows that S 3 Q η 689λ+ρ Q η 689 +c Q a +a+α ρ Q <δ λ Q δ/ η 689 c Q a +a+α k q λ δ q k k 4η 689 By 7 and th fact that η 689 < /4, w hav

22 454 JF Morgnbssr / Journal of Numbr Thory S 3 τ q λ Q η 689λ+ρ Q η 689 +c Q a +a+α ρ Q <δ λ Q δ/ η 689 c Q a +a+α + λ δ log Q 689 4η 689 = τ q λ Q λ/ 4η 689 log Q 689+ρ Q η 689 +c Q a +a+α ρ Q <δ λ Q δ 4η 689 log Q 689 c Q a +a+α Sinc w obtain c Q a + a + α c Q π log Q < 4η 689 log Q 689, 9 Thus w gt s 6, 8, 9 and 30 S 3 τ q λ Q λ/ c Q a +a+α +ρ Q 30 S ν τ q λ Q λ c Q a +a+α +ρ Q Exactly th sam way as in [6, Lmma 0], on can show that τ q λ λ ωq τ q, whr ωq dnots th numbr of distinct prim divisors of q Hnc w obtain compar also with and 5 By 3 w hav S ν ωq+ Q ν c Q a +a+α +3ρ log 689 Q S Q ν γρ/ + Q ν νκ ρ/ + Q ν/ max S r,ν,ρ / r q ρ Q ν γρ/ + Q ν νκ ρ/ + ν ωq/+ Q ν c Q a +a+α +3/ρ log 689 Q Until now, w only usd that ρ ν/3 If w impos th condition thn w hav c Q ρ ν min a + a + α κ, γ + 3log 689 Q + γ, S ν ωq/+ Q ν γρ/ W st κ c q,κ = min + γ, c Q γ + 3log 689 Q, and choos ρ := ν c q,κ a + a + α Ifρ, thn w just hav shown th dsird rsult cf 0 If ρ <, thn th stimat 0 holds trivially and w ar don

23 JF Morgnbssr / Journal of Numbr Thory Proof of Thorm and Thorm 3 Thorm and Thorm 3 can b dducd from Thorm Bfor w prov thm, w show two auxiliary lmmas concrning congruncs in th Gaussian intgrs and th κ-z[i] squnc Lmma Lt q = a ± i, a Z + and m Z satisfying m a + a + Furthrmorltb Z, = + i Z[i] and st d = m, q Thnwhav ± a + b mod m if and onlyif b mod d Hr th choic of th sign dpnds on th sign for q = a ± i Proof First, w prov th claim that ± a + b mod m if and only if a + ba + mod m 3 W hav that a +, m a +, a + a + = a +,a + + = Thus th lft-hand sid of 3 is quivalnt to ± a + a + ba + mod m Sinc m a + a +, this is quivalnt to th right-hand sid of 3 and th claim is shown Nxt w show that whn δ is dfind by δ = m, q Indd,wcanwrit m δ = m, a + a + m,q q = = m, q m, q W obtain th last inquality from th fact that This can b asily shown using th idntity m δ = d, 3 {, if a is odd, q, q = + i, othrwis m, q m, q m, q q, q = a ± i, a i = a ± i, i = a ± i, + i As a last prparation not that q = i ia + Now w can prov th statd rsult Lt us assum that b mod d Using 3, this is quivalnt to b mod m/δ, whr δ is dfind as abov Sinc m/δ, ia + /δ = thisisquivalntto + i ia + b ia + mod m, i, it is quivalnt to ± a + + i a + b + i ba + mod m

24 456 JF Morgnbssr / Journal of Numbr Thory Rcall that d, b Z Thus, w gt by our first claim that th last statmnt is quivalnt to ± a + b mod m Lmma 3 Lt b, d Z[i] and D N N N b a κ-z[i] squnc Thn w hav # { D N : b mod d } = #D N d #{ R d : b mod d } + O d N κ Proof W dfin for Z[i] th numbrs ω to b if b mod d and 0 othrwis Thn w can writ # { D N : b mod d } = d + d ω +r D N r R d d ω D N ω +r r R d D N whr w choos R d to b a complt rsidu systm modulo d with r d for all r R d Thfirst trm is qual to, It rmains to stimat th rror trm W hav d ω D N #D N d #{ R d : b mod d } ω +r r R d D N ω r R d D N D N ω +r { # D N r + D N } r Rd r R d r N κ d 3 N κ Proof of Thorm W hav { # D N : s q } b mod g = j sq b g g D N 0 j<g St d = g, a + a + From [9, Corollary 3] follows that s q = s q + i ± a + mod d If w put g = g d, J ={kg : 0 k < d}, J ={0,,g }\ J ={kg + r: 0 k < d, r < g },thn w hav for j = kg J, j g s q = k d s q = k d ± a +

25 JF Morgnbssr / Journal of Numbr Thory Hnc, j sq b g g D N j J = g D k d ± a + b N 0 k< d = d g #{ = + i D N : ± a + b mod d} Sinc d, q = g, q, q = g, q = d, w gt by Lmma and Lmma 3 that th last quantity is th sam as d g #{ D N : } b mod d = #D N d g d Q b, d + O q,g N κ Using th sam considrations as in Lmma, w s that d/ d = Th rmaining sum which coms from th st J can b tratd with Thorm and on finally obtains Thorm s th proof of Thorm 3 in [6] for dtails Proof of Thorm 3 If α Q, thn th squnc αs q Z[i] taks modulo only a finit numbr of valus and is thrfor not uniformly distributd modulo If in rturn α R \ Q, thnforvry h Z with h 0whava + a + hα R \ Q and according to Thorm tak D N = Ξ N, th statmnt follows from Wyl s critrion s [6, Thorm 9] 6 Proof of Thorm 4 As alrady indicatd in Sction, w hav { # D N : s q = } k = 0 Sα αk dα, 33 whr Sα := D N αs q Wdfin IN,k := { D N : k mod q } W hav cf [9, Corollary 3] s q R ± a + I mod a + a + and s q mod q With Lmma w can charactri IN, k in two diffrnt ways, namly, IN,k = { D N : s q k mod a + a + } = { D N : R ± a + I k mod a + a + } Furthrmor, w st RN, k := #IN, k It follows from Lmma 3 that RN,k = #D N a + a + #{ R q : k mod q } + O q N / 34

26 458 JF Morgnbssr / Journal of Numbr Thory Proposition Lt q = a ± i, whr a Thn w hav for vry non-ngativ intgr k, αs q = RN,k αμ Q log Q N π α σq log Q N + O α log log N 9 35 IN,k uniformly for ral α with α log log Nlog N / 6 Proof of Proposition W know from Lmma 7 that thr xists a constant c 0, such that th Gaussian intgr has lss than or qual to L := log Q N + c digits, whnvr D N Lt us fix som intgr k In what follows, w will prov a slightly diffrnt rsult, namly that n In,k αs q n = RN,kαμ Q L π α σ Q L + O α 4 L + O α log L 9 36 uniformly for ral α with α log LL / This implis th dsird rsult Now w want to translat 36 into a probabilistic languag Lt us considr th st IN, k Ifw assum that vry numbr in this st is qually likly, thn th function which assigns ach numbr its j-th digit D j,n := ε j is a random variabl Hnc, th sum-of-digits function s q can also b intrprtd as a random variabl S N = s q := j L ε j Using this modl, formula 36 is quivalnt to th rlation ϕ t := E its N Lμ Q /LσQ / = t / t 4 log L9 + O + O t L L 37 that is uniform for t 4πσ Q log L st α = t/πσ Q L / As a first stp w truncat th sum-of-digits function St L = # { j Z: log L 9 j L log L 9} = L log L 9 + O, and T N = log L 9 j L log L 9 ε j n Furthrmor, lt ϕ t b th charactristic function of T N L μ Q /L σ Q /, i, ϕ t := E itt N L μ Q /L σ Q /

27 JF Morgnbssr / Journal of Numbr Thory Lmma 4 W hav uniformly for all ral t ϕ t ϕ t = O t log L9 Proof Th proof of this lmma follows almost litrally th proof of [4, Lmma 4] and w omit it hr On only has to not that by our dfinition of L w hav L L log L 9 and S N T N log L 9 L Lt Z j b a squnc of indpndnt random variabls with rang {0,,,Q } and uniform probability distribution St T N := log L 9 j L log L 9 Z j Thn w hav ET N = L μ Q nd latr on and VT N = L σ Q Nxt w stat a wll-known proprty which w will Lmma 5 W hav E wt N L μ Q /L σq / = w / w 4 + O L that is uniform for w L 4 Proof S for xampl Lmma 4 of [4], which is a slight variant of this statmnt th proof is actually th sam In particular, for th charactristic function of th normalid random variabl T N th rlation ϕ 3 t := E itt N L μ Q /L σq / = t / t 4 + O L holds whnvr t L 4 In what follows, w will show that T N is a good approximation of th truncatd sum-of-digits function To do so, w hav to compar ϕ and ϕ 3 Proposition W hav uniformly for ral t with t πσ Q log L, ϕ t ϕ 3 t = O t L, whr th implid constant is absolut Combining Lmma 4, Lmma 5 and Proposition w obtain 37 which in turn provs Proposition

28 460 JF Morgnbssr / Journal of Numbr Thory Erdős Turán Koksma inquality and stimats of xponntial sums In ordr to prov Proposition w will also nd th concpt of th discrpancy of squncs If x,,x M ar points in th -dimnsional ral vctor spac R and H an arbitrary positiv intgr, thn th two-dimnsional vrsion of th Erdős Turán Koksma inquality says that th discrpancy D M of ths points satisfy D M H + + M h x rh l M, 0< h H l= whr h Z and rh = max, h max, h For an accurat dfinition of th discrpancy and th statmnt abov s [6, Sction ] As w will s latr on, this yilds to quadratic xponntial sums Gittnbrgr and Thuswaldnr showd in [8, Chaptr ], that A tr B Nlog N σ, <N whnvr A, B = and log N σ B N log N σ This stimat is too wak for proving Proposition and it is only shown for discs with radius N In what follows, w us th van dr Corput inquality Lmma 6 to trat such xponntial sums This nabls us to improv th rror trm on th on hand and it allows us to considr /-Z[i] squncs instad of discs on th othr hand Lmma 6 Lt A, B Z[i] with A, B = and lt D N N N b a /-Z[i] squnc Furthrmor lt c > 0 b ral and σ Z + such that Q 4clog log Nσ B N Q 4clog log Nσ Thn w hav D N A tr B NQ clog log Nσ log N, whr th implid constant is absolut Proof Lt us dnot th considrd sum as S Using Lmma 6 with R = B / yilds N S B N/ B / r A B / + tr + r / + N / B / B r B /,+r Ξ N A / tr B r + N / B /,+r Ξ N r R B Hnc w hav to stimat linar xponntial sums ovr th Gaussian intgrs Th important proprty is that w ar summing ovr rctangls with sid lngth smallr than N,+r Ξ N A N N tr B r min N, s, s, s s,

29 JF Morgnbssr / Journal of Numbr Thory whr s + is = r A W thrfor gt B S N/ B / min N, r R B / N N R A B r, I A B r, R A A r I B B r + N / B / If A, B =, thn Ar also runs through a complt rsidu systm modulo B Employing Lmma 0 yilds S N/ B / N + B N / log B + B log B / + N / B / log B N B / + N / B / Using th bounds on B brings th dsird rsult If A, B wwrit A B = A B with A, B = and w can do similar calculations as abov Sinc B and B ar comparabl not that A, B = w obtain th sam rsult in this cas, too 6 Joint distribution of th summands of T N and T N Th main stp of proving Proposition is th comparison of th momnts of T N and T N Thrfor w nd som information on th joint distribution of thir summands Proposition 3 Lt d L and j, j,, j d and l,l,,l d intgrs with log L 9 j < j < < j d L log L 9 and l,l,,l d {0,,,Q } Thn w hav uniformly RN,k #{ n IN,k: ε j n = l,,ε } jd n = l d = Q d + O d c r + d c 3r c log L 9 + log L+c 3rd log L9, whr c,c and c 3 ar positiv constants and r > 0 is an arbitrary intgr W adopt th notion of Gittnbrgr and Thuswaldnr [8, Chaptr 3] and dfin th fundamntal domain of th bas-q rprsntation systm by F = { C: = } ε j q j, ε j N j= Evry complx numbr can b rprsntd as = α 0 + α q with uniqu ral numbrs α 0 and α Thus, th mapping ϕ : C R, = α 0 + α q α 0,α is wll dfind and it is calld th ϕ-mbdding of F in R Wst F := ϕ F = { R : = } E j ε j, ε j ϕn, j=

30 46 JF Morgnbssr / Journal of Numbr Thory with 0 a E = a Not that ϕq = Eϕ Ifm N, w will hav to dal with th domain containing all th numbrs whos fractional parts start with th digit m W dnot th mbddd vrsion by F m = E F + ϕm Sinc this rgion has a rathr complicatd shap, w hav to approximat it Thrfor w us th following lmma which is provd in [8, Lmma 3] Lmma 7 For all m N and all r N thr xists an axially paralll tub P r,m with th following proprtis: i F m P r,m for all r N ii λ P r,m = Oμ r /Q r iii P r,m consists of Oμ k axially paralll rctangls, ach of which has Lbsgu masur OQ r Th constant μ satisfis < μ < Q In th proof of th lmma Gittnbrgr and Thuswaldnr constructd a polygon Π r,m with axparalll sids such that P r,m = { R : Π r,m c q r}, whr c is an absolut constant that can b chosn For th rmaining part of this sction w fix to ach pair r,m th polygon Π r,m, th corrsponding tub P r,m and dnot by I r,m th st of all points insid Π r,m Wdfin f m x, y = / / Ψ m x + x, y + y dx dy, / / whr = c q r and, if x, y I r,m, Ψ m x, y = /, if x, y Π r,m, 0, othrwis Th function f m is a so-calld Urysohn function which quals for x, y I r,m \ P r,m,0forx, y R \ I r,m P r,m and is an intrpolation of ths valus in btwn Th nxt lmma givs stimats for th Fourir cofficints of this function and can b found in [8, Lmma 3 and Lmma 33] Lmma 8 Lt f m x, y = n,n Z c n,n n x + n y b th Fourir xpansion of f m Thn for th Fourir cofficints c n,n w gt th stimats

31 JF Morgnbssr / Journal of Numbr Thory μ r c n,n = O n n,n 0, n μ r c n,0 = O n 0, c 0,n = O c 0,0 = Q n μ r n n 0, Furthrmor w hav for n,n 0 that c n,n = 0 if q qn n Bfor w start proving Proposition 3, w nd an auxiliary lmma W st { F j = # IN,k: ϕ q j+ P r,m mod Z } m N Lmma 9 W hav uniformly for log L 9 j L log L 9, whr c is a positiv constant μ r RN,k F j + μ r c log L 9, Q Proof From Lmma 7 it follows that w can subdivid ach tub P r,m into a family of Oμ r rctangls which hav Lbsgu masur OQ r such that w hav whr F j G m is dfind by RN,k F j { F j G m = # IN,k: ϕ F j G m, 38 RN, k m N G m P r,m q j+ G m mod Z }, and th scond sum in 38 runs ovr all rctangls G m in which w subdivid P r,m In what follows w show that thr xists a positiv constant c such that th discrpancy D of th squnc ϕ q j+ whr IN, k is boundd by D c log L 9 This thn implis RN,k F jg m λ G m + D Q r + c log L 9, and th rsult of th lmma follows s Lmma 7 This lads us to th Erdős Turán Koksma inquality W hav D H + + rh h ϕ RN, k q j+ 39 0< h H IN,k

32 464 JF Morgnbssr / Journal of Numbr Thory It is asy to s that τ := tr, T a trq = Xϕ with X := a a Thn w hav h ϕ = q j+ hx τ = h q j+ tr q + h, j+ q j whr h, h T := hx Not furthrmor, that R + a + I = tr ia + Using th dfinition of IN, k on asily obtains for th innr sum in 39, W st IN,k h ϕ q j+ = a +a+ a + a + l=0 h tr n D N kl a + a + q j+ + h q j + A B = h q + h l ia + + j+ q j a + a +, l ia + a + a + with A, B = Thn q j /H B q j+4 not, that h, h Q 3/ If w st H = Q 3 log L9, w hav W thn us Lmma 6 to obtain D H + H + Q 3 log L9 B N Q 3 log L9 Q c log L 9, N RN,k Q 6 log L9 log N 0 h H N RN,k Q 6 log L9 log Nlog H rh whr c is a suitabl positiv constant not that N/RN, k, s 34 This concluds th proof of Lmma 9 Proof of Proposition 3 W st t l,j = d h= f lh ϕ, q j h+ whr l = l,,l d and j = j,, j d Th following fundamntal rlation allows us to us our just obtaind rsults W hav

33 JF Morgnbssr / Journal of Numbr Thory #{ IN,k: ε j = l,,ε } jd = l d F j + + F jd μ r drn,k + μ r c log L 9 Q IN,k t l,j Thus it rmains to study th sum IN,k t l,j It is asy to s that t l,j = M M T M d h= μ h ϕ, q j h+ whr M ={M = μ,,μ d : μ h = m h,m h with m h,m h Z; h =,,d}, and T M = d h= c m h,m h Hnc w can writ IN,k t l,j = M M T M IN,k d h= μ h ϕ q j h+ If M = 0, thn T M = Q d by Lmma 8 If M = μ,,μ d 0 such that thr xists an intgr h with q qm h m h thn T M = 0 again by Lmma 8 In all othr cass w hav using th sam notation as in th proof of Lmma 9 d h= μ h ϕ = q j h+ d h= = tr i μ h X τ q j h+ d h= qm h m h q j, h+ whr th choic of th sign dpnds on th sign of q = a ± i Ifwst A B = i d h= qm h m h, q j h+ whr A, B =, w s that q j + B q j d+ not, that q qm h m h Furthrmor w hav IN,k t l,j = a + a + = a + a + D N t l,j a +a+ r=0 T M M M D N a +a+ r=0 r R + a + I k a + a + rk a + a + A tr r ia + + B a + a +

34 466 JF Morgnbssr / Journal of Numbr Thory If w writ th innr sum as D N tr A B, w s that B B B sinc q, a + a + = Thus w hav Q log L9 B N log L9 Q and Lmma 6 yilds IN,k t l,j RN,k Q d + O NQ 4 log L9 log N M 0 T M It is asy to s that M T M μr d μ r Q r d Finally, w st c = logμ/q, c 3 = logμq and w obtain th dsird rsult Lmma 0 W hav for d L, T N L μ d Q T N L μ d Q E = E + EN, d, r, L σ Q L σ Q whr EN, d, r = O Q L / d d c r + d c 3r c log L 9 + log L+c 3rd 4 log L9, and th constants ar th sam as in Proposition 3 Proof Th proof works xactly as th proof of [4, Lmma 46] Thus w omit it 63 Proof of Proposition Th final stps in th proof of Proposition ar vry similar to th considrations in [4, Sction 43] W giv hr only a rough outlin Using Taylor s thorm w hav for vry intgr D > 0, E it X E ity = it d EX d EY d d! d<d t D + O E X D E Y D + t D E Y D, D! D! whr X = T x L μ Q /L σ Q / and Y = T x L μ Q /L σ Q / W choos D = log L 3 and assum without loss of gnrality that D is vn and mploy Lmma 0 with r = log L 5 Ashort calculation shows that for our choic of D and r w obtain uniformly in d D EX d EY d log L4 /L Hnc w gt rcall that t 4πσ Q log L d D t d EX d EY d t d! d D 4πσ Q log L d d! log L4 L t 4πσ Q log L log L4 L t L

35 JF Morgnbssr / Journal of Numbr Thory Th sam computations thn in [4, Sction 43] show that W finally obtain EY D D! D D/ D/ D / 6 Final stps in th proof of Thorm 4 t D D! E Y D t πσ Q D log L D D D/ D/ D t /L / In this subsction w collct alrady provd rsults to show Thorm 4 It is straightforward cf [4] and w thrfor giv only a rough outlin First, w us th priodicity of th intgrand to obtain Sα αk dα = a + a + a +a+ S k α αk dα, 0 a +a+ whr S k α = αs q IN,k Now w considr th last intgral sparatly whthr α is small or big, namly a +a+ a +a+ = α log log Nlog N / + log log Nlog N / < α /a +a+ 40 Th scond intgral whr α is big, can b boundd abov using Thorm Indd, this thorm also implis th uppr bound S k α log N ωq/+ N c q,κ a +a+α, 4 and w obtain for th scond intgral in 40 th stimat S k α αk dα log N ωq/+ NN c q,κ a +a+ log log N log N N log N Sinc #D N N, this xprssion is boundd by th rror trm statd in th thorm For th uppr bound of th first intgral in 40, w us Proposition W hav

36 468 JF Morgnbssr / Journal of Numbr Thory S k α αk dα α log log Nlog N / = RN,k α μq log Q N k π α σq log Q N dα + RN,k α log log Nlog N / O α log log N 9 dα α log log Nlog N / Nxt, w us th substitution α = t/πσ Q log Q N / and obtain RN, k πσ Q log Q N / it k t / dt + RN,k O log N / t / dt t log log N log log N + RN,k O log N RN, k = k / + O log log N / log log N + O, πσ Q log Q N log N whr k = k μ Q log Q N σ Q log Q N / SincthfirstO-trm is boundd by th scond on, th rsult follows Acknowldgmnts Th author would lik to thank Michal Drmota and Joël Rivat for many inspiring discussions Furthrmor, I would lik to thank Thomas Stoll for proofrading and hlpful suggstions, and Christoph Habrl and Christian Stindr for rmarks on κ-z[i] squncs Rfrncs [] Nadr L Bassily, Imr Kátai, Distribution of th valus of q-additiv functions on polynomial squncs, Acta Math Hungar [] Bruc C Brndt, Ronald J Evans, Knnth S Williams, Gauss and Jacobi Sums, Canad Math Soc Sr Monogr Adv Txts, John Wily & Sons Inc/Wily Intrscinc Publication, Nw York, 998 [3] Michal Drmota, Ptr J Grabnr, Pirr Liardt, Block additiv functions on th Gaussian intgrs, Acta Arith [4] Michal Drmota, Christian Mauduit, Joël Rivat, Prims with an avrag sum of digits, Compos Math [5] Michal Drmota, Joël Rivat, Thomas Stoll, Th sum of digits of prims in Z[i], Monatsh Math [6] Michal Drmota, Robrt F Tichy, Squncs, Discrpancis and Applications, Lctur Nots in Math, vol 65, Springr- Vrlag, Brlin, 997 [7] Alksandr O Glfond, Sur ls nombrs qui ont ds propriétés additivs t multiplicativs donnés, Acta Arith [8] Brnhard Gittnbrgr, Jörg M Thuswaldnr, Asymptotic normality of b-additiv functions on polynomial squncs in th Gaussian numbr fild, J Numbr Thory [9] Ptr J Grabnr, Ptr Kirschnhofr, Hlmut Prodingr, Th sum-of-digits function for complx bass, J Lond Math Soc [0] Ptr J Grabnr, Pirr Liardt, Harmonic proprtis of th sum-of-digits function for complx bass, Acta Arith [] Sidny W Graham, Grigori Kolsnik, Van dr Corput s Mthod of Exponntial Sums, London Math Soc Lctur Not Sr, vol 6, Cambridg Univrsity Prss, Cambridg, 99 [] Hnryk Iwanic, Almost-prims rprsntd by quadratic polynomials, Invnt Math