On lower bounds for the L2L2-discrepancy

The L2L2-discrepancy measures the irregularity of the distribution of a finite point set. In this note, we prove lower bounds for the L2L2-discrepancy of arbitrary NN-point sets. Our main focus is on the two-dimensional case. Asymptotic upper and lower estimates of the L2L2-discrepancy in dimension 2 are well known, and are of the sharp order logN. Nevertheless, the gap in the constants between the best-known lower and upper bounds is unsatisfactorily large for a two-dimensional problem. Our lower bound improves upon this situation considerably. The main method is an adaption of Roth’s method, using the Fourier coefficients of the discrepancy function with respect to the Haar basis. We obtain the same improvement in the quotient of lower and upper bounds in the general dd-dimensional case. Our lower bounds are also valid for the weighted discrepancy.