On Sums of Fractional Parts {nα+γ}

Abstract

We use the continued fraction expansion ofαto obtain a simple, explicit formula for the sumCm(α, γ)=∑1⩽k⩽m({kα+γ}−12)whenαis irrational. From this we deduce a number of elementary bounds on the growth and behaviour ofCm(α, γ). In particular, we show that as m varies the extent of the fluctuations in size can be determined almost entirely from the non-homogeneous continued fraction expansion ofγwith respect toα. These sums are closely related to the discrepancy of the sequence ({nα}); we state a related explicit formula that yields similar bounds for the discrepancy. Sums of this form also occur in a lattice point problem of Hardy and Littlewood.