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Sums of Reciprocals of Fractional Parts and Multiplicative Diophantine Approximation
About this Title
Victor Beresnevich, Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom, Alan Haynes, Department of Mathematics, University of Houston, Texas and Sanju Velani, Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 263, Number 1276
ISBNs: 978-1-4704-4095-4 (print); 978-1-4704-5660-3 (online)
DOI: https://doi.org/10.1090/memo/1276
Published electronically: February 24, 2020
Keywords: Multiplicative Diophantine approximation,
Ostrowski expansion,
uniform distribution
MSC: Primary 11K60, 11J71, 11A55, 11J83, 11J20, 11J70, 11K38, 11J54, 11K06, 11K50
Table of Contents
1. Problems and main results
2. Developing techniques and establishing the main results
Abstract
There are two main interrelated goals of this paper. Firstly we investigate the sums \begin{equation*} S_N(\alpha ,\gamma ):=\sum _{n=1}^N\frac {1}{n\|n\alpha -\gamma \|} \end{equation*} and \begin{equation*} R_N(\alpha ,\gamma ):=\sum _{n=1}^N\frac {1}{\|n\alpha -\gamma \|}\,, \end{equation*} where $\alpha$ and $\gamma$ are real parameters and $\|\cdot \|$ is the distance to the nearest integer. Our theorems improve upon previous results of W. M. Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of \begin{equation*} \{1\le n\le N:\|n\alpha -\gamma \|<\varepsilon \} \, , \end{equation*} valid for all sufficiently large $N$ and all sufficiently small $\varepsilon$. This first strand of the work is motivated by applications to multiplicative Diophantine approximation, which are also considered. In particular, we obtain complete Khintchine type results for multiplicative simultaneous Diophantine approximation on fibers in $\mathbb {R}^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.- Dzmitry Badziahin and Sanju Velani, Multiplicatively badly approximable numbers and generalised Cantor sets, Adv. Math. 228 (2011), no. 5, 2766–2796. MR 2838058, DOI 10.1016/j.aim.2011.06.041
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