2 CHAPTER 1 . UNIFOR M DISTRIBUTIO N Then th e discrepancy o f th e sequenc e i s (4) D*(N) = su p \D(N-a)\. a€[0,l] Thus th e condition s (a)-(c ) abov e ar e equivalen t t o (d) D*(N) = o(N) as N - oo. The discrepancy D*(N) i s not invariant unde r translation, whic h is regrettable since T is a homogeneous space . Th e usua l metho d use d t o overcome thi s defec t involves counting the number Z(N a , (3) o f n fo r whic h u n e [a , /?] (mo d 1) . W e put (5) D(N a , (3) = Z(N a , f3) - (0 - a)N, assuming tha t a /3 a + 1 , and the n w e set (6) D(N) = sup\D(N a,0)\. a, P Equivalently, w e may pu t (7) D(N) = sup D(N a) - infD{N a), a a and i t i s evident tha t (8) D*(N) D{N) 2D*(AT). This new discrepancy D i s translation-invariant, bu t i t is more complicated tha n necessary. T o construc t a n alternativ e approach , le t s(x) denot e th e saw-toot h function (9) s(x) = { { n 10 x £ {x} - 1/ 2 x i Z , o x eZ. Here {x} denote s th e fractiona l par t o f £, {x} = x [x\. Le t N (10) 6{N ] a) = J2 s (un-(*)- n=l That is , 6(N a) = / T s(x - a) dZ(N\ a) . W e pu t (11) fi(N)=sup|fi(JV a)|. a. If x 7 ^ 0,x =fi a (mo d 1) , then x f0 ,(x ) = a + s( x a) s(x) wher e x § denote s the characteristic functio n o f the set 8 . W e set x = u n , su m over n, an d subtrac t Na t o se e tha t D*{N a)=6(N',a)-6{N 0) at point s o f continuity . Thu s 6(N a) differ s fro m D*(N a) b y a n additiv e constant. I n th e cas e o f D*(N\a), th e additiv e constan t i s chose n s o tha t
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