The Inverse of the Star-discrepancy Depends Linearly on the Dimension

We study bounds on the classical ∗-discrepancy and on its inverse. Let n∞(d, ε) be the inverse of the ∗-discrepancy, i.e., the minimal number of points in dimension d with the ∗-discrepancy at most ε. We prove that n∞(d, ε) depends linearly on d and at most quadratically on ε−1. We present three upper bounds on n∞(d, ε), all of them are based on probabilistic arguments and therefore they are non-constructive. The linear in d upper bound directly follows from deep results of the theory of empirical processes but it contains an unknown multiplicative factor. Two other upper bounds are without unknown factors but do not yield the linear (in d) upper bound. One upper bound is based on an average case analysis for the Lp-star discrepancy and our numerical results seem to indicate that it gives the best estimates for specific values of d and ε. We also present two lower bounds on n∞(d, ε). For lower bounds, we allow arbitrary coefficients in the discrepancy formula. We prove that n∞(d, ε) must be of order d log ε−1 and, roughly, of order dλε−(1−λ) for any λ ∈ (0, 1).