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Discrepancy and uniform distribution of sequences

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Discrepancy and uniform distribution of sequences COMPOSITIO MATHEMATICA EDMUNDHLAWKA Discrepancy and uniform distribution of sequences Compositio Mathematica, tome 16 (1964), p. 83-91. <http://www.numdam.org/item?id=CM_1964__16__83_0> © Foundation Compositio Mathematica, 1964, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 83 Discrepancy and uniform distribution of sequences* by Edmund Hlawka First 1 want to repeat some définitions : If we have a point x in R8, we call his coordinates pj(x) (j = 1, ... s). Two points x, x’ are congruent modulo 1, if x - x’ has integral coordinates. Now if we are only interested in properties modulo 1, we can always suppose that x is in the unit cube E = Es : 0 Pi 1 (j = 1, ... s ) in other words we consider only the fractional part {x} of X. Now we consider first infinite sequence (0: zi, x2, ... of points or more general 0) : x11; x x22; ... ; x ... xNN; ... all lying in E8. Now we consider an intervall Q : ocj ~ pj 03B2j (j --- 1, ... s) in Es. Let,yQ (x) be the characteristic function of Q. Then N(Q, a» = 03A3Nk=1~Q (aek) is the number of points aen of the given sequence with n N lying in Q. If we have a double sequence then we define N(Q, co) = 03A3Nk=1 ~Q(xNk). The sequence 03C9 is called uniformly distributed modulo 1 if for each interval Q of E8, limN~~ N(Q, ro)/N exists and is equal to the volume of Q : ft(Q) = E ~Q(x)dx. This means that [1] Sometimes it is better to write it in another form. Let Q be an arbitrary interval 03B1j ~ Pi 03B2j in RI,

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