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For a sequence xI,...,xN of points in [0,1] and a sequence p,. PN (PI + P2 + + PN = 1) of nonnegative numbers, define the distribution function g(x) = x E PA. -yk < V Let Tp be an increasing function on [0,1] and 9p(0) = 0. The main result of the paper is F(DN) < J 9T( g(x)l) dx < 9p(DN), where DN is the supremum norm of g on [0,1] and F is the antiderivative of q with F(O) = 0. This result generalizes and improves an estimate of Niederreiter [1] for the L2 discrepancy of the sequence xI,.X.,XN. Applying the above inequality we also obtain a new criterion for uniform distribution modulo one.

Generalization of two results of the theory of uniform distribution