标题： 倒数之和和三个距离定理 显示英文标题

标题： Sums of reciprocals and the three distance theorem

摘要： 在本文中，我们研究等差数列倒数的和对一取模的情况，即 $\{n\alpha-\gamma\}^{-1}$的和，其中 $\alpha$和 $\gamma$是实参数，而 $\{\,\cdot\,\}$是实数的小数部分。 这些和的界已经长期与各种应用相关联进行了研究。 在本文中，我们开发了一种替代技术来获得这些和的上界，并得到了新的高效且完全明确的结果。 该技术使用所谓的三距离定理。 显示英文摘要

摘要： 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.