Lethbridge Number Theory and Combinatorics Seminar: Nicol Leong

We investigate the sums of reciprocals to an arithmetic progression taken modulo one, that is sums of fractional parts 1/{na−y}, where a,y are real parameters. Such upper bounds are known in the literature, with many applications in number theory. For instance, they related to exponential sums and are also used in counting lattice points in polygons. We present an alternate and novel method to obtain new efficient and fully explicit results. The technique uses the Three Distance Theorem and the theory of continued fractions. This is based on joint work with Victor Beresnevich. If time permits, we will also briefly discuss Poissonian pair correlation of gaps.