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.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: continued fractions, sums of fractional parts, discrepancy
      Subjects: 11-xx Number theory > 11Jxx Diophantine approximation, transcendental number theory
      11-xx Number theory > 11Axx Elementary number theory
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 20 Nov 2003
      Last Modified: 21 Apr 2010 11:13

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