The $L sup 2$ Discrepancies of the Hammersley and Zaremba Sequences in $0,1 sup 2$ for an Arbitrary Radix

Abstract

Useful theoretical formulae are presented for measuring, in a quadratic-mean sense, the extent to which a class of important sequences is imperfectly distributed in the unit square. Previous results of Halton and Zaremba are generalized for sequences based on an arbitrary radix. The new discrepancy formulae are exact and much easier to analyze and evaluate than previously known versions. The formulae have direct application in providing significantly improved error-bounds in the Quasi-Monte Carlo numerical integration of difficult functions.