Let AN be an N-point set in the unit square and consider the discrepancy function
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where x = (x1, x2) ∈ [0,1;]2,
and |[0, x)]| denotes the Lebesgue measure of the rectangle. We give various refinements of a well-known result of Schmidt [Irregularities of distribution. VII. Acta Arith. 21 (1972), 45–50] on the L∞ norm of DN. We show that necessarily
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025579300000930/resource/name/S0025579300000930_eqnU2.gif?pub-status=live)
The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we have
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025579300000930/resource/name/S0025579300000930_eqnU3.gif?pub-status=live)
whenever N = 2n for some positive integer n. This estimate depends upon variants of the Chang–Wilson–Wolff inequality [S.-Y. A. Chang, J. M. Wilson and T. H.Wolff, Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv.60(2) (1985), 217–246]. We also provide similar estimates for the BMO norm of DN.