Sums of Reciprocals of Fractional Parts II
arXiv:2304.02566
Abstract
We prove an estimate for the number of lattice points lying in certain non-convex Euclidean domains of interest in Diophantine approximation. As an application, we generalise a result of Kruse (1964) concerning the almost sure order of magnitude of sums of reciprocals of fractional parts and solve a conjecture posed by Beresnevich, Haynes, and Velani. The methods are based both on the geometry of numbers and on probability theory.
- Appendix by Michael Björklund, Reynold Fregoli, and Alexander Gorodnik - This is the NEW VERSION of a paper entitled Sums of Reciprocals of Fractional Parts over Aligned Boxes (identifier arXiv:2101.11094) by the same author. The previous version will not be published