Recent Progress in Improvement of Extreme Discrepancy and Star Discrepancy of One-Dimensional Sequences

Abstract

In this communication, we report on recent progress in improvement of extreme discrepancy and star discrepancy of one-dimensional sequences. Namely, we present a permutation of “Babylonian” sequences in base 60, which improves the best known results for star discrepancy obtained by Henri Faure in 1981 [Bull. Soc. Math. France, 109, 143–182 (1981)], and a permutation of sequences in base 84, which improves the best known results for extreme discrepancy obtained by Henri Faure in 1992 [J. Numb. Theory, 42, 47–56 (1992)]. Our best result for star discrepancy in base 60 is 32209/(35400log 60)≈0.222223 (Faure’s best result in base 12 is 1919/(3454log 12)≈0.223585); our best result for extreme discrepancy in base 84 is 130/(83log 84)≈0.353494 (Faure’s best result in base 36 is 23/(35log 6)≈0.366758).