An Explicit Formula for the L₂-Discrepancy of (nα)-Sequences

Abstract

Assume that α is an irrational number with continued fraction expansion [a₀;a₁, . . .] and convergents ,n= 0, 1 . . . . Every positive integer N has a unique expansion N=b₀q₀+b₁q₁+ . . . +b _m q _m , where the digits b _i are nonnegative integers with b₀<a₁,b _i ≤a _{i +1} and such that b _i =a _{i +1} implies b _{i −1}=0, the so-called Ostrowski expansion of N to base α. On the other hand let c_{[0, x )} be the characteristic function of the half-open interval [0,x) and let

be the L₂-discrepancy of the sequence (nα) mod 1, where {y} denotes the fractional part of the real number y. In this paper, we give an explicit formula for D*⁽²⁾_N(α) entirely in terms of the digits b₀, . . . ,b _m . This formula enables one to compute the L₂-discrepancy in at most O(log⁴N) steps, where the O-constant does not depend on α, while in the classical formulae for the L₂-discrepancy N²/2 calculation steps are necessary. The fastest algorithm so far was found by S. Heinrich [3] and works (for all sequences) in O(N log N) steps.