On the discrepancy of circular sequences of reals

Abstract

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