# On Sums of Fractional Parts $\{ n\alpha + \gamma\}$

Pinner, Christopher (1995) On Sums of Fractional Parts $\{ n\alpha + \gamma\}$. [Preprint]

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We use the continued fraction expansion of $\alpha$ to obtain a simple, explicit, formula for the sum $C_{m}(\alpha,\gamma)=\sum_{1\leq k\leq m} \left( \left\{ k\alpha +\gamma \right\} -\frac{1}{2} \right)$ when $\alpha$ is irrational. From this we deduce a number of elementary bounds on the growth and behaviour of $C_{m}(\alpha,\gamma)$. In particular we show tha t as $m$ varies the extent of the fluctuations in size can be determined almost entirely from the non-homogeneous continued fraction expansion of $\gamma$ with respect to $\alpha$. These sums are closely related to the discrepancy of the sequence $(\{ n\alpha \})$; 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.