Sums of Fractional Parts of Integer Multiples of an Irrational Original Research Article

Let α be a positive irrational real number, and let Cα(n) = ∑1 ≤ k ≤ n ({kα} − image), n ≥ 1, where {x} denotes the fractional part of x. We give an explicit formula for Cα(n) in terms of the simple continued fraction for α, and use this formula to give simple proofs of several results of A. Ostrowski, G. H. Hardy and J.E. Littlewood, and V. T. Sós. We also show that there exist positive constants dA such that if α = [a0, a1, a2, ...] and (1/t) ∑1 ≤ j ≤ taj ≤ A holds for infinitely many t, then Cα(x) > dA log x and Cα(x) < −dA log x each hold for infinitely many x.