On the discrepancy of circular sequences of reals

In this paper we study a refined measure of the discrepancy of sequences of real numbers in [0,1][0,1] on a circle C   of circumference 1. Specifically, for a sequence x=(x1,x2,…)x=(x1,x2,…) in [0,1][0,1], define the discrepancy  D(x)D(x) of x byD(x)=infn≥1⁡infm≥1⁡n‖xm−xm+n‖ where ‖xi−xj‖=min⁡{|xi−xj|,1−|xi−xj|}‖xi−xj‖=min⁡{|xi−xj|,1−|xi−xj|} is the distance between xixi and xjxj on C  . We show that supx⁡D(x)≤3−52 and that this bound is achieved, strengthening a conjecture of D.J. Newman.