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Exponential Squared Integrability of the Discrepancy Function in Two Dimensions

Part of: Harmonic analysis in one variable Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  21 December 2009

Dmitriy Bilyk ,
Michael T. Lacey ,
Ioannis Parissis  and
Armen Vagharshakyan
Dmitriy Bilyk
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, U.S.A., E-mail: bilyk@math.ias.edu
Michael T. Lacey
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, U.S.A., E-mail: lacey@math.gatech.edu
Ioannis Parissis
Affiliation:
Institutionen för Matematik, Kungliga Tekniska Högskolan, SE 100 44, Stockholm, Sweden, E-mail: ioannis.parissis@gmail.com
Armen Vagharshakyan
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A., E-mail: armenv@math.gatech.edu
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Abstract

Let AN be an N-point set in the unit square and consider the discrepancy function

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

The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we have

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.

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy 42A05: Trigonometric polynomials, inequalities, extremal problems
Type
Research Article
Information
Mathematika , Volume 55 , Issue 1-2 , December 2009 , pp. 1 - 27
Copyright
Copyright © University College London 2009

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