The Exponent of Discrepancy Is at Least 1.0669

LetP⊂[0, 1]^dbe ann-point set and letw: P→[0, ∞) be a weight function withw(P)=∑_z∈P w(z)=1. TheL₂-discrepancy of the weighted set (P, w) is defined as theL₂-average ofD(x)=vol(B_x)−w(P∩B_x) overx∈[0, 1]^d, where vol(B_x) is the volume of thed-dimensional intervalB_x=∏^d_k=1 [0, x_k). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd⩾1 and allε>0 there exists a weighted set of at mostKε^−ppoints in [0, 1]^dwithL₂-discrepancy at mostε, whereK=K(p) is a suitable number independent ofεandd. Wasilkowski and Woźniakowski proved thatp*⩽1.4779, by combining known bounds for the error of numerical integration and using their relation toL₂-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*⩾1.04882, and by a slight modification of the proof we getp*⩾1.0669. Determiningp* exactly seems to be quite a difficult problem.