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Abstract
There are two main interrelated goals of this paper. Firstly we investigate the sums
$S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\n\alpha\gamma\}$
and
$R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\n\alpha\gamma\},$
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 $\{1\le n\le N:\n\alpha\gamma\<\ve\} \, ,$ valid for all sufficiently large $N$ and all sufficiently small $\ve$.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 $\R^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.
$S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\n\alpha\gamma\}$
and
$R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\n\alpha\gamma\},$
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 $\{1\le n\le N:\n\alpha\gamma\<\ve\} \, ,$ valid for all sufficiently large $N$ and all sufficiently small $\ve$.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 $\R^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.
Original language  English 

Pages (fromto)  1124 
Journal  Memoirs of the American Mathematical Society 
Volume  263 
Issue number  1276 
DOIs  
Publication status  Published  24 Feb 2020 
Bibliographical note
© 2020, American Mathematical Society This is an authorproduced version of the published paper. Uploaded in accordance with the publisher’s selfarchiving policy. Further copying may not be permitted; contact the publisher for details.Profiles
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research