8 CHAPTE R 1 . UNIFOR M DISTRIBUTIO N Moreover, s~(k) = -l/(2nik) fo r k ^ 0, so that XJ(k) = (e(ka)~e(kp))/(2wik). Hence ] sin 7r/c(/3 a) i lx,(*)l = irk ^min(/?-Q'i) when k ^ 0. O n combinin g thes e inequalitie s w e find tha t (22) |S+ (/0 |^+n^_a,_L) for 0 |fc| if, an d henc e tha t K D { N^)^ i+ 2^j^^n{p-a,^))\U N (k)\. Since a lower boun d fo r D(N a,/3) ma y b e similarly derive d usin g S K (x), w e arrive at the followin g result . THEOREM 1 . Let {un}^=1 be a sequence of points ofT, let 3 = [a,/? ] be an arc ofT with a (3 a -f1, and let D(N\ a , /?) 6 e the discrepancy given in (5). Then (23) | D ( J V a , / 3 ) | ^ + 2 ^ (s ^ I + m i n ( / J - a , ^ ) ) | f «(*«„ ) /or an y positive integers N and K. Since j ^ x T ^ ^ 2k + ^ r ^:5 w e obtain th e followin g familia r estimate . (24) COROLLARY 1.1 . (Th e Erdos-Tura n inequality ) For any positive integer K, K - , i N + W*K^ +3 £*1X* U » A =l ' n = l The advantag e o f the Theore m ove r th e Corollar y i s that th e coefficient s on the righ t han d sid e ar e smaller i f the interva l 3 is short. Thu s th e Theorem not onl y yield s a n estimate o f discrepancy, bu t also allow s u s to estimate the maximum ga p in the sequenc e u n , in terms of the exponentia l sum s U]y(k). COROLLARY 1.2 . Suppose that K is a positive integer chosen so that K , A T (25) Yl\Yle^kUr^ fc=l'n=l AT/10 . Then every arc 3 = [a , /?] of T of length (3—a -^y contains at least \N(f3 a) of the points u n , I n N.
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