The Supremum Norm of the Discrepancy Function: Recent Results and Connections

A great challenge in the analysis of the discrepancy function DN is to obtain universal lower bounds on the L∞ norm of DN in dimensions d ≥ 3. It follows from the L2 bound of Klaus Roth that ‖DN‖∞ ≥ ‖DN‖2 & (logN)(d−1)/2. It is conjectured that the L∞ bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d = 2. Partial improvements of the Roth exponent (d−1)/2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.