Sums of reciprocals and the three distance theorem

11 Dec 2017  ·  Beresnevich Victor, Leong Nicol ·

In this paper we investigate the sums of reciprocals to an arithmetic progression taken modulo one, that is sums of $\{n\alpha-\gamma\}^{-1}$, where $\alpha$ and $\gamma$ are real parameters and $\{\,\cdot\,\}$ is the fractional part of a real number. Bounds for these sums have been studied for a long while in connection with various applications... In this paper we develop an alternative technique for obtaining upper bounds for the sums and obtain new efficient and fully explicit results. The technique uses the so-called three distance theorem. read more

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Number Theory