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] on a circle C of circumference 1. Specifically, for a sequence x = (x(1), x(2), ...) in [0,1], define the discrepancy D(x) of x by D(x) = inf(n >= 1) inf(m >= 1) n parallel to x(m) - x(m+n)parallel to where parallel to x(i) - x(j)parallel to = min {vertical bar x(i) - x(j)vertical bar, 1 - vertical bar x(i) - x(j)vertical bar} is the distance between x(i) and x(i) on C. We show that sup(x) D(x) <= 3-root 5/2 and that this bound is achieved, strengthening a conjecture of D.J. Newman. (C) 2016 Elsevier Inc. All rights reserved.