Determination of [nθ] by its Sequence of*Differences

Abstract

For any real numberθletwhere [x] denotes the greatest integer not exceedingx.A method is given for computingf_θfrom its first few terms. A similar method is given for computing the characteristic functiong_θ(n)of[nθ]. The given methods converge rapidly, and generalize previous results of Bernoulli, Markorf, and Stolarsky. Note that either of the sequencesf_θand g_θdetermines the sequence[n_θ] (n = 1, 2,…).