WARING S PROBLEM RESTRICTED BY A SYSTEM OF SUM OF DIGITS CONGRUENCES

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1 WARING S PROBLEM RESTRICTED BY A SYSTEM OF SUM OF DIGITS CONGRUENCES OLIVER PFEIFFER AND JÖRG M. THUSWALDNER Abstract. Th aim of th prsnt papr is to gnraiz arir work by Thuswadnr and Tichy on Waring s Probm with digita rstrictions to systms of digita rstrictions. Lt s q n b th q-adic sum of digits function and t d, s, a,, q N. Thn for s > d 2 og d + og og d + O thr xists N N such that ach intgr N N has a rprsntation of th form N = x d + + x d s whr s q x i a mod i s and L. Th rsut, togthr with an asymptotic formua of th numbr of this rprsntations, wi b shown with th hp of th circ mthod togthr with xponntia sum stimats.. Notation Lt N, Z and R dnot th st of positiv intgrs, intgrs and ra numbrs, rspctivy. A st of th shap {n Z a n b} wi b cad intrva of intgrs. Th notations z for xp2πiz, x for th gratst intgr ss than or qua to x R, and x for th smast intgr gratr than or qua to x wi b usd frqunty. For th sak of shortnss, w ar going to mak xtnsiv us of vctor and matrix notation throughout this papr. For xamp, if v,..., v d is a finit coction of indxd numbrs, thn v = v,,v d wi dnot th corrsponding vctor. Furthrmor w wi us th notations fx = O gx as w as fx gx to xprss that fx c gx for som positiv constant c and a sufficinty arg x R. A function f is said to b compty q-additiv, if for any p, r, t N with r < q t th proprty fp q t +r = fp+fr hods. Th cassica xamp of a compty q-additiv function is th th q-adic sum of digits function s q which assigns to ach positiv intgr n th sum s q n = c + + c r of digits in its uniqu q-adic rprsntation n = c + c q + + c r q r. This function wi pay a prominnt ro throughout th papr. Dat: August 7, Mathmatics Subjct Cassification. P5, L5, A63, P55. Ky words and phrass. Asymptotic basis, Waring s Probm, sum of digits function. Rsarch supportd by th Austrian Scinc Foundation FWF Projct Nr. S83.

2 2 OLIVER PFEIFFER AND JÖRG M. THUSWALDNER 2. Main rsuts and priminaris A fundamnta probm in additiv numbr thory is to dcid whthr a givn st A N is a basis of N, that is, if ach N N admits a rprsntation of th form N = x + + x s with x,,x s A. W ca s N th ordr of th basis. If a rprsntation of this shap ony xists if N is sufficinty arg w ca A an asymptotic basis of N. In Waring s Probm th st A is to b takn A = A d = {n d n N} d N fixd. This probm and variants of it hav bn studid xtnsivy. For dtais and rfrncs w rfr for xamp to Hua [5], Nathanson [8], Vaughan [] or Vaughan and Wooy []. In [9], Thuswadnr and Tichy invstigatd th numbr of rprsntations 2. N = x d + + x d s, whr th intgrs x i hav bn additionay rstrictd by sum of digits congruncs of th typ s qi x i a i mod m i for givn a i, q i, and m i, i s cf. [9, Thorm 3.]. As consqunc, thy dducd that th st {n d s q n a mod m} forms an asymptotic basis of ordr 2 d + cf. [9, Thorm 3.2]. In th prsnt papr w go on stp furthr and gnraiz this work to systms of digita rstrictions. In particuar, for givn positiv intgr N N w considr th numbr r d,s,a,m N of rprsntations 2. whr ach x i, i s, simutanousy obys a systm of L sum of digits congruncs s q x i a mod m,. s ql x i a L mod m L. W ar going to provid an asymptotic formua for r d,s,a,m N from which th fact that th corrsponding rstrictd st forms an asymptotic basis wi foow. W wi us th abbrviation s q n a mod m if s q n a mod hods for L, and dnot th st of a intgrs that fufi this condition by U a,m = {n N s q n a mod m}. Sts of that kind hav aso bn studid with diffrnt stups at first by Gfond [3] and subsqunt authors as for xamp by Bsinau [], Mauduit and Sárközy [7] and Kim [6]. Our main rsut can b summarizd as foows. Thorm 2.. Lt d, s N and a,, q N with 2, q 2, gcdq, = for L and gcdq,q k = for < k L. If r d,s,a,m N dnots th numbr of rprsntations of N in th form N = x d + + x d s x,,x s U a,m,

3 WARING S PROBLEM RESTRICTED BY SUM OF DIGITS 3 thn for s > d 2 og d + og og d + O th impid constant is absout thr xists a positiv constant δ such that r d,s,a,m N = M ssnγ + d s s s Γ N d + ON s d δ, d whr M = L. Th impid constant dpnds ony on d, s, L and m. S is an arithmtic function for which thr xist positiv constants < c < c 2 dpnding ony on d and s such that c < SN < c 2. This impis that A d,a,m = {n d s q n a mod m} forms an asymptotic basis of ordr d 2 og d + og og d + O of N. In ordr to stabish this thorm, w wi nd th foowing highr corration rsut for s q n provd by Thuswadnr and Tichy in th aformntiond papr [9]. To formuat it, t fn,k dnot th d-th itratd diffrnc oprator appid to an arithmtic function f with diffrncs k,..., k d, i.., fn,k = fn + k fn, fn,k,,k d = fn,k,,k d,kd d 2. Proposition 2. [9], Thorm 3.3. Lt d, m, h, q and N b positiv intgrs with m 2, q 2 and m hq, and t dd + 2 pd,q = 2 q + 2d + 5. Lt I,, I d, J b intrvas of intgrs with N I j, J N for j d. Thn th stimat h m d sq n,k 2 I I d J 2 N η k I n J k d I d hods with η = /m 2 q pd,q >. With th hp of this rsut w wi driv th foowing stimat which is crucia in th proof of Thorm 2.. Thorm 2.2. Lt d,, h, q and N b positiv intgrs with 2, q 2 for L, gcdq,q k = for < k L and h q for at ast on L. Thn th stimat N θn d + L h s q n N γ hods uniformy in θ [, with γ = η/6dl 2, whr D = 2 d, η = minη > and η = /m 2 qpd,q > with pd,q as in Proposition 2..

4 4 OLIVER PFEIFFER AND JÖRG M. THUSWALDNER Kim [6, Proposition 2] provd a vrsion of this rsut with θ =, i.., whr th trm θn d is missing. Mor prcisy, h showd that undr th sam conditions as in Thorm 2.2 for a positiv intgrs N N L h f n N δ, whr δ = /2m 2 q 3 L 2 with m = max, q = maxq, and th impid constant dpnds ony on q and L. In fact, Kim s rsut is vn mor gnra sinc it admits arbitrary compty q -additiv functions f instad of th s q. On th othr hand, Thuswadnr and Tichy [9, Thorm 3.4] providd th cas L = of our rsut. Thy showd that if d,m,h,q and N ar positiv intgrs with m 2, q 2 and m hq, thn th stimat N θn d + h m s qn = N ε hods uniformy in θ [, with ε = η2 d+ and η as in Proposition 2.. Comparing this with th spcia cas L = of Thorm 2.2, thir saving ε is obviousy bttr than our γ, which is inhrnty du to th diffrnt mthod foowing Kim [6] appid. Thorm 2.2 constituts a gnraization of both of ths rsuts. Not that it vn rmains vaid if th trm θn d is rpacd by an arbitrary poynomia in n of dgr d. In ordr to stabish Thorm 2.2 w hav to adapt th proof of [6, Proposition 2] to our mor sophisticatd situation. This wi ad to xponntia sums which can b stimatd with hp of Proposition 2.. Thorm 2. wi thn foow from Thorm 2.2 by an appication of th circ mthod. Nxt w ar going to provid two priminary mmata. Th foowing is a gnraization of [6, Lmma 6]. Lmma 2.. Lt f b a compty q-additiv function. Thn fn,k = d fr,k for a positiv k,, k d and n with n r mod q t, whr r < q t k k d. Proof. It is asy to s that th d-th itratd diffrnc oprator can b writtn xpicity as fn,k = d I f n + k i. i I I {,,d}

5 WARING S PROBLEM RESTRICTED BY SUM OF DIGITS 5 Lt n = p q t + r. Sinc r + i I k i < q t for any sction of th subst of indics I, w can xpoit th q-additivity of f and obtain fn,k = d I f pq t + r + k i = I {,,d} i I = fp d I + d I f r + k i = fr,k. I {,,d} I {,,d} i I } {{ } = Th foowing inquaity is a variant of [4, Lmma 2.7] which is itsf an itration of th ordinary Wy-van dr Corput inquaity. Lmma 2.2. Lt D = 2 d and K. Thn th inquaity N 2.2 ϕn D 32 D N D K K K + ND K d hods for any arithmtic function ϕn. k = k d = N k k d ϕn,k Proof. W ony giv a sktch of th asy proof. Lt A j b dfind rcursivy by A = and A j = A 2 j j for j 2. Starting from th ordinary Wy-van dr Corput inquaity cf. [6, Lmma 4], w obtain N ϕn D A d N D d j= 2 2D 3 2d j j= K 2d j + N D 22D 2 K d K k = K k d = N k k d ϕn,k by induction and itratd appication of Cauchy-Schwarz s inquaity. Again by induction on can show that A d 2 2D d which is in turn 2 2D 2. Now d 2 2D 3 2d j max j 2 d 2D 3 2d j = d 22D 3 K 2d j min j K 2d j K. Sinc w ar ony intrstd in a rsut simiar to [4, Lmma 2.7], w gnrousy stimat th nominator by 2 3D 3. This yids inquaity 2.2. j= 3. Proof of Thorm 2.2 In this sction w ar going to driv Thorm 2.2 from Proposition 2. and w do this by foowing th proof of [6, Proposition 2]. Lt q = maxq. W hav to invstigat th probm ony for N q 3L, bcaus for N q 3L th stimat hods triviay. St K := N /3L q 2, and t Q = q such that 3. 2 K K 2 q Q K 2

6 6 OLIVER PFEIFFER AND JÖRG M. THUSWALDNER for L. This can b achivd by choosing = 2 og K/ og q. W start from th itratd Wy-van dr Corput inquaity 2.2 with L ϕn = θn d h + s q n, so that th ft hand sid of inquaity 2.2 is th D-th powr of th xponntia sum w want to stimat. Sinc θn d,k = θd! k k d is constant with rspct to n, and by th inarity of, L h ϕn,k = θd! k k d + sq n,k, w hav for fixd k,..., k d 3.2 N k k d ϕn,k = θd! k k d } {{ } = N k k d L h sq n,k. At this point w distinguish in which rsidu cass n mod Q is. To accompish this, t th sts R and R b dfind by Furthrmor, for r R t R = {r Z L r Q for L}, R = {r Z L r Q d K for L}. P r = {n Z n r mod Q}, whr n r mod Q mans that n r mod Q for L. With hp of ths sts w rwrit th sum undr th rightmost moduus of quation 3.2 in th foowing way: 3.3 N k k d L h sq n,k = r R N k k d To th first sum w can now appy Lmma 2. to obtain r R N k k d n P r L h sq n,k = n P r L h sq n,k = = r R + = L N k kd h sq r,k = r R r R n P r r R\R. r R\R.

7 WARING S PROBLEM RESTRICTED BY SUM OF DIGITS 7 Substituting this back into quation 3.3 ads to 3.4 N k k d + r R\R L h sq n,k = L r R N k k d n P r L h sq n,k L N k kd h sq r,k + n P r h sq r,k = = + 2. Now w argu aong th sam ins as Kim [6, p. 33]. Sinc th Q ar pairwis coprim as powrs of th q, th systm of congruncs that dfins th st P r is quivant to a sing congrunc mod L Q by th Chins Rmaindr Thorm, and thrfor {n < N n r mod Q} = N L Q + O. Hnc for th first sum w gt Q = r = Q L r L = sq r,k N m L Q + O = h = N Q Q r = h sq r,k + O L Q. Sinc L Q K 2L N/K by 3., th rror trm in th ast quation is of ordr ON/K. Th scond sum is boundd triviay by 2 2 R\R N L Q + O 2dqL N K + O K Q, whr w usd th stimat from [6, p. 33] for R\R. By th sam argumnt as bfor th fina rror trm is of ordr ON/K. Substituting ths two stimats again back into quation 3.4 w obtain N k k d L h sq n,k = N Q Q r = h sq r,k N + O. K

8 8 OLIVER PFEIFFER AND JÖRG M. THUSWALDNER Now from th itratd Wy-van dr Corput inquaity 2.2 w gt 3.5 N θn d + By stting L 32 D N D K + ND K d h D s q n 3 = K d K k = K k = K Q Q k d = r = K Q Q k d = r = and appying Hödr s inquaity to this trm w obtain 3 K d L+ K d K k = K k d = K Q Q r = K K k = h sq r,k N D + O. K d+ h sq r,k h sq r,k L+ L+ K k d = Q Q r = h sq r,k 2 L+, whr th ast inquaity is vaid sinc th vau undr th moduus is at most. Now th ast trm is a product of sums of th typ bing stimatd in Proposition 2.. By assumption, for at ast on L w hav h q, say. Sinc K Q K 2 for any L by 3., or in othr trms Q K Q, this mans that w can in fact appy Proposition 2. to gt K K K k = K k d = Q Q r = h sq r,k 2 Q η K η with η > as in Thorm 2.2. Stting η = minη > yids K η K η, and this is N η/3l bcaus K = N /3L /2N /3L. Estimating th rmaining factors corrsponding to triviay by, w obtain N η/3ll+ N η/6l2 3 and, obsrving inquaity 3.5, finay N θn d + L h D s q n 32 D N D m +O N D N D + O = 3 K K d+ = O N D η/6l2 + O N D /3L + O N D d+/3l. Stting γ = η/6dl 2, th thr summands on th right hand sid ar a O N D γ, and taking th D-th root yids Thorm 2.2.

9 WARING S PROBLEM RESTRICTED BY SUM OF DIGITS 9 h = 4. Appication of th circ mthod In this sction w ar going to prov Thorm 2.. W wi foow th ins of Thuswadnr and Tichy [9, Sction 9] to a grat xtnt. Lt P = N /d. Using th idntity m h { if s q n a mod m, sq n a = othrwis, mntary mthods of additiv numbr thory yid 4. r d,s,a,m N = for th numbr r d,s,a,m N in qustion, whr Fθ = P θn n U a,m = M P m h = P d = θn d m L h L = Fθ s θndθ θn d + m h = L h sq n a = h sq n a and M = L. With th hp of th L s-matrics H = h i L, i s and M = L, i s w can xprss Fθ s as Fθ s = M s s = M s P m i= n i = h i = s H<M i= n i = m L P h Li = θn d i + θn d i + whr th matrix inquaity is to b undrstood as foows: L L h i sq n i a = h i sq n i a, H < M = {H Z L s h i < for i s and L}. Insrting this xprssion into 4. ads to r d,s,a,m N = M s H<M i= s P L θn d h i i + sq n i a θndθ. m i= n i = } {{ } I H Concrning th intgras I H with H w obsrv that s 4.2 I H S i θ dθ sup Si θ s 2t max θ,i j S j θ 2t dθ

10 OLIVER PFEIFFER AND JÖRG M. THUSWALDNER for any t with s > 2t, whr P S i θ = θn d + L h i s q n ar th ssntia sums to which w can appy Thorm 2.2. Writing intgra 4.2 as in th proof of th cassica Lmma of Hua cf. [, Lmma 2.5] S j θ 2t dθ = n,,n 2t <P n d ++nd t =nd t+ ++nd 2t L h j t sq n k s q n t+k and appying Ford [2, Equation 5.4], this intgra can obviousy b furthr boundd from abov by S j θ 2t dθ { } n,,n 2t < P n d + + n d t = n d t+ + + n d 2t P 2t d for th currnty bst known owr bound t > /2d 2 og d + og og d + O whr th impid constant is absout. Substituting th ast stimat togthr with th on from Thorm 2.2 back into quation 4.2 w obtain k= I H P γ s 2t P 2t d = P s d δ with δ = γs 2t >, which hods for s > 2t > d 2 og d + og og d + O. Th intgra I is w-known from th ordinary Waring s Probm and can b vauatd using th Hardy-Littwood asymptotic formua, which aso hods for s > d 2 og d + og og d + O cf. Nathanson [8, Thorm 5.7], Vaughan and Wooy [, p. ] and Ford [2]. Putting P = N /d back into th stimats compts th proof. Rfrncs [] J. Bésinau. Indépndanc statistiqu d nsmbs iés à a fonction somm ds chiffrs. Acta Arith., 2:4 46, 972. [2] K. B. Ford. Nw stimats for man vaus of Wy sums. Intrnat. Math. Rs. Notics, 3:55 7, 995. [3] A. O. G fond. Sur s nombrs qui ont ds propriétés additivs t mutipicativs donnés. Acta Arith., 3: , 968. [4] S. W. Graham and G. Kosnik. van dr Corput s mthod of xponntia sums, voum 26 of London Mathmatica Socity Lctur Not Sris. Cambridg Univrsity Prss, Cambridg, 99. [5] L. K. Hua. Additiv thory of prim numbrs. Transations of Mathmatica Monographs, Vo. 3. Amrican Mathmatica Socity, Providnc, R.I., 965. [6] D.-H. Kim. On th joint distribution of q-additiv functions in rsidu casss. J. Numbr Thory, 742:37 336, 999. [7] C. Mauduit and A. Sárközy. On th arithmtic structur of sts charactrizd by sum of digits proprtis. J. Numbr Thory, 6:25 38, 996. [8] M. B. Nathanson. Additiv numbr thory, voum 64 of Graduat Txts in Mathmatics. Springr- Vrag, Nw York, 996. Th cassica bass.

11 WARING S PROBLEM RESTRICTED BY SUM OF DIGITS [9] J. M. Thuswadnr and R. F. Tichy. Waring s probm with digita rstrictions. Isra J. Math., 49:37 344, 25. Probabiity in mathmatics. [] R. C. Vaughan. Th Hardy-Littwood mthod, voum 25 of Cambridg Tracts in Mathmatics. Cambridg Univrsity Prss, Cambridg, scond dition, 997. [] R. C. Vaughan and T. D. Wooy. Waring s probm: a survy. In Numbr thory for th minnium, III Urbana, IL, 2, pags A K Ptrs, Natick, MA, 22. Dpartmnt of Mathmatics and Statistics, Univrsity of Lobn, Franz-Josf-Strass 8, A-87 Lobn, AUSTRIA E-mai addrss: oivr.pfiffr@uniobn.ac.at Dpartmnt of Mathmatics and Statistics, Univrsity of Lobn, Franz-Josf-Strass 8, A-87 Lobn, AUSTRIA E-mai addrss: jorg.thuswadnr@uniobn.ac.at