Mathematics. - On a prblem in the thery f unifrm distributin. By P. ERDÖS and P. TURÁN. 11. Ommunicated by Prf. J. G. VAN DER CORPUT.) Cmmunicated at the meeting f September 5, 1948.) 9. Af ter this first reductin f ur prblem we transfrm it in the fllwing way. Let fr the plynmial V'z) defined in 8.4) Then M;= max 1V'z)l= max lv'i z)l= max 1 ii1-ze-il'~)i 9.1) Izl=6 Izl=6 Izl=6 ~=I max lg lv'i z)1 = lg M;. 1 z 1=6. 9.) Since lg V'd z) is regular fr I z I = {} we have by the classica 1 therem {}. f HADAMARD-BREL-CARATHEODORY that fr I z I :;;;;,smce 10gV'1 0) =0, i.e. Ilg '1'1 z) 1-= lg M; r denting I e-ki'f~ n 1'=1 by Sk. 1 k~: ~ S-k 1-= lg M;. I z 1-= ~. 9.3) Then CAUCHY's estimatin gives I I -I I -= 10g M; st - s_. = k ~ )' = k! r lg M; <! r lg M; =! r g* :, })' 9.4) Owing t the trivial inequality I Sk I :;;;; n the inequality 9. 4) is restrictive nly fr thse k's fr which k -= lg g* ~' })..... 9. 5) lg-; Hence the prf f Therem 11 is reduced t the questin whether r nt the inequalities 9.4) with the restrictin 9. 5) invlve equidistributin
163 f the cp/s md n. In ther wrds we reduced the prf f Therem Il t the prf f Therem III with Since m 1jJ k) n 4 m ) m ) n 4 ) m m. ~~I k < f/ n'1) -:& 1 + lg + u n. d) -:&. m. < arid m < u. :'1)!) m ~ 1 +!) lg ; ~ -== :::::: n ~)m:::::: n < 3n - u n'1) t'1 - y' g. n'1) lg u n. ij) n 4 n m +1 < lg -:&. lg u n'1) we btain - anticipating Therem III - that fr every 0 :s; a < P :s; n we have I.z 1 - P - a n I < 5 C lg ~. n "~I'~ ~ timdn " t'1 lg u n'1) i.e. Therem Il will be prved. 10. Befre turning t the prf f Therem lil we sketch the crres~ pnding reasning fr Therem 1. In this case - as we remarked in 1) - the general case can be reduced t the case wh en all the rts lie n the unit circle. Then in 9. ) M~ is replaced by maxl1jjz) 1= M. Applying Izl=1 the therem f HADAMARD-BREL-GARATHEODORY t the interir circle I z I = e. where we determine e suitable later. we btain resp. -== lg M Ilg 1jJI z) I = -1--. -e CAUCHY's estimatin gives f Zk S-k I < lg M I k=1 k 1 -e.... 10. 1) -== lg M 1 ISkl =IS-kl=k -1--' k' -e e
164 Chsing e = 1 - k! 1 I Sk I -= 0 k lg M ~ k = I,,... ~....... 10. ) This gives accrding t the crllary f Therem 111 an errr term On" /s ) nly. Hence Therem I seems t he much deeper. This seems t justify the use f mre difficuit analytica I tls in the prf f l). 11. Fr the prf f therem 111 we need sme simple auxiliary cnsideratins. Let Ohviusly n 1,. [mj+ 1 sin t R 1 510". [TJ+l)4 [mj+l)3 "[T] +1) sin t 1 )4 dt.... 11.1) R > 4: t dt = 4: Si; y dy > 0 11. ) > ~ ; tf Si; yr dy > Cl m 3 where Cl and later C. dente numerical cnstants. Further [ m 4 [ J \4 ] +1 m +1) n sin t,. sin - t R=f dt-=n4f dt< t 510 " 0 11. 3) 1. Let a he a parameter suhjected nly t the restrictin n==-a==- ml~1 1.1)
165 and let [ m] + 1 4 a -x nmx,a)= kj sin----t -x sm " dt.. 1.) We have als m +1 [ ] sin t-x) t t-x sin-- 4 dt.. 1.3) and sinee the integrand f 1. ) is an even funetin f t x [m] + 1 sin t nmx,a)= kj x-a sm " dt.... 1. i) Sinee generally. k + 1 ) sm- -y. =k+ 1)+k esy+k-l)csy+... l csky, sm 1L we btain at nee that the integrand f 1. ), i.e. :n m x, a) itself, is a trignmetrie plynmial f rder :s; m :nm x, a) = a a) + av a) cs vx + bv a) sin vx).. 1.5) I~"~m 13. We need sme infrmatin abut the eefficients in 1.5). Evidently, using 1.3),,. a,. a a) = 1 :nj :nm x,a)dx= ~Rj'd1 000 a = :n 1 R J dt. R = a n' m +1 [ ] sin t-x) t-x sin-- 4 dx= 13. 1)
Further using 1. ) 166 n n IJ 1 0 Jà1l: m a,.a)=- 1I:mx.a)csvxdx=-- -à sinvxdx= 11: ' 1I:V X i.e. n =--- IJ. SlDVX similarly 1I:vR n la,.a)l-= 1I:~RJ + [m]+1 1 [m] + 1 sin a-x) sin x. a-x + x sm-- sin - [ m] + I 1 sin a-x). a-x sin-- [m]+ 1 sin x dx=-. X 11: V SlD-! Ib,.a)I-=~ 11: V 1 -= v-= m. + I-=v-=m: 1 dx. 13.)... 13.3) 14. We need ais sme infrinatin abut the shape f the graph f 1I: m x, a). The definitin f Rand representatin 1.) give immediately fr every reai x -= 1I:m x. a) -= 1....... 14. I) We cn si der 1I:m x, a) in the interval -= -= m+ I =x=a- m+ I 14.) this has a meaning wing t 1. 1) ). Using the representatin 1. ) and the 'estimatin 11. 3) we have in the range 14. ) 1I:m x. a) > -1-3 J I sin t I + [-'i] [m] + 1 C m t 1 dt.= 14.3)
167 Further, fr a < x ~ i:n we have, using the estimatin 11. ) and the r~presentatin 1.4) :n m x, a) < _1-3J-:'" cim x-a [m] + 1 sin t 510" 14.4) Finally, fr t :n ~ X ~ :n we have frm the estimatin 11. ) and the 'TC representatin 1. 4), since x - a ~ i:n - :n = x 'TCmx, a) <f [ ~]+1 sin t 510 '" 14.5) 3", [;] + 1 )1 sin y J = C6 Y dg < m 'TC-X»3 "'-x 15. Nw we are ging t prve the fllwing Lemma. Assuming 4.1) and 4.) we have fr the number N f the cp; 5 lying in an arbitrary interval f leng th m 1~ 1 with m > 0 the C7 inequality Prf. n m 1jJ v») N<cs -+-1 + -... 15.1) m ~=I v Withut lss f generality we may suppse that ur interval is We cnsider the plynmial :n m x, y) wh ere 14 y= m + l'.... 15.3)
168 Replacing x in m nm x. r) = a r) + I a~ r) cs vx + b~ r) sin vx) ~=I by cp!. CP... cpn and summing we btain wing t 13. 1) nrm n m n I~ n m CPI. r) = n n + ~/d\ a~ r) I~\ cs VCPI + ~~ b~ r) I~ sin v CPI. In the interval 15. ) cnditin 14. ) is satisfied i.e. frm 14. 3) and the nn~negativity f nm x. y) we have n 7 n m n C3 N -=: I~\ n m CPI. y) -=: -; m + 1 + ~~ I a~ y) 1 + 1 by r) J)~!t ey1'f11 Applying 4.1). 13.) and 13.3) webtain further C3 N < 1-. _n_ + ~ i tp v). n m+1 n ~=\ v Q. e.d. 16. Nw we turn t the prf f therem 111. Let d be given satisfying 10 -=:d-=: m+1= =n.. 16. 1) and cnsider the plynmial nm x. d). Replacing x in d m n m x. d) = n + ':~I a y d) cs v X + b" d) sin v x) by CPl CP... cpn and summing we btain n d m n m n j/dlnm qjj d) = n n +,,~ a" d)j:; cs vqjj +,,~ b.> d) j~1 sin vcpj Arguing as befre we btain n d 4 m tp v) I n m qjj. d) > - n- - I -... 16.) j=\ n n ~ =I v Nw denting the number f cp;s in 0:;;;; x ~ d by Nd)he cntributin f these cp;s is. wing t nm x. d) ~ 1. -=: N d)...... 16.3) T btain an upper estimatin fr the cntributin f the ther cpv's we cnstruct successive cntiguus intervals f length m 1~ 1 each starting frm x = d and cvering the interval d ~ x ~ t n. The cntributins f the cpv in the interval Dk d + k m 1~ 1 -=: x -=: d + k + 1) m 1~ 1
169 is wing t the lemma which is, by 14. 4) Hence summing ver k < C8 -+ 1 +.' -- max n m x, d) n m VI v)) m ~=I v x Dk n m tij < C -- +.' V)) C _T_ max 5 < 8 m+l ~=I v xedkmx-d))3 n m VI v)) 1 <C9 m+l + 3:1-V-. P'.. 16.4) The cntributin E the gjv lying in the remaining interval 3 n < x :s; n we can estimate similarly. Cmbining 16. ), 16. 3) and 16. 4) we btain d n m VI v)) Nd»- ' n-cii -+1 +.' -... n m ~=I v 16. 5) Obviusly the same estimatin hlds Er the nu mb er Nc, c + d) E the lfjv's Er which c :s; gjv :s; c + d md n. The restrictin d :s; n is bviusly unnecessary, ie we replace Cll by Cll = C1' d T btain the upper estimatin E Nc, c + d) - n ~ we have. due t merely t apply 16. 5) twice. Ncl C + d) = n-no, c)-nc + d, n) Q.e.d. University f Syracuse lnstitute fr AdlJ'anced Study.