4 CHAPTER 1 . UNIFOR M DISTRIBUTIO N On takin g F(x) = e(-kx) i n (13) , w e see a t onc e tha t |[//v(A ) | 7r\k\D(N) for k 0. Her e th e constan t ca n b e improve d b y takin g a little mor e care . Express th e comple x numbe r [/#(& ) i n pola r coordinates , UN{k) = pe(0) wher e p=\UN(k)\. The n N \UN(k)\ = e(-6)UN(k) = ]T e(-few n - 0) . n=l Since thi s su m i s real , i t suffices t o conside r th e rea l part s o f th e summands . Thus N \UN(k)\ = ^cos27r{ku n + 6). n=l On takin g F(x) = cos27r(/ca -f 0) i n (13) , we deduce tha t (14) \U N (k)\ 2\k\D(N) for al l k ^ 0 . I f w e tak e un = n/k then [/jv(fc ) = N and D(JV ) - N/k. Thu s the uppe r boun d i s within a constant facto r o f being best possible . O n the othe r hand, fo r an y given sequence un ther e are not many values of k for which \U^{k)\ is this large . T o show thi s w e first not e tha t 6(N k) = -U N (k)s(k) = l ^§- for k 7^ 0. Sinc e 6(0) = 0 , an d sinc e UN(—^) = UN(k), i t follow s b y Parseval' s identity tha t (15) f l^^=2» 2 /Wa)ada. Hence i n particular , K Y,\UN(k)\2^K2D{N)\ fc=i so that ther e ar e a t mos t boundedl y man y /c , 1 fc K for whic h |C/TV(/C) | is of the order o f KD(N). (I n the abov e we have employed Vinogradov's C notation, which is synonymous with the Big-0 notation popularized by Hardy and Landau . Precisely, w e writ e / C g o r / = O(g) i f ther e i s a n absolut e constan t A such that |/ | Ag uniforml y fo r al l value s o f the fre e variable s unde r consideration . The constan t A is referre d t o a s th e implicit constant.) Th e relation s (14 ) an d (15) provid e goo d quantitativ e passag e fro m (d ) t o (b) . The usua l derivatio n o f (a ) fro m (b ) involve s th e existenc e o f trigonometri c polynomials T~(x),T + (x) suc h tha t T-(x)xM{x)T+{x)
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