Sums of reciprocals and the three distance theorem

2017/12/11 by Victor Beresnevich, Beresnevich, Victor, Nicol En Ci Leong +1
Mathematics · #11J13 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1712.03758

openalex publication_date 2017/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we investigate the sums of reciprocals to an arithmetic progression taken modulo one, that is sums of \nα-γ\^-1, where α and γ are real parameters and \ ⋅ \ 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.

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