On lower bounds for the L_2-discrepancy

Details On lower bounds for the L_2-discrepancy On lower bounds for the L_2-discrepancy

2010-10-07

arxiv.org/abs/1010.1950v3

Journal of Complexity 27 (2011), 127-132

Mathematics

Numerical Analysis

Scientific paper

10.1016/j.jco.2010.11.002

The L_2-discrepancy measures the irregularity of the distribution of a finite point set. In this note we prove lower bounds for the L_2 discrepancy of arbitrary N-point sets. Our main focus is on the two-dimensional case. Asymptotic upper and lower estimates of the L_2-discrepancy in dimension 2 are well-known and are of the sharp order sqrt(log N). Nevertheless the gap in the constants between the best known lower and upper bounds is unsatisfactory large for a two-dimensional problem. Our lower bound improves upon this situation considerably. The main method is an adaption of the method of K. F. Roth using the Fourier coefficients of the discrepancy function with respect to the Haar basis.

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