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There are two main interrelated goals of this paper. Firstly we investigatethe sums \[S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}~~~\text{and}~~~R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|}\,, \] where$\alpha$ and $\gamma$ are real parameters and $\|\cdot\|$ is the distance tothe nearest integer. Our theorems improve upon previous results of W. M.Schmidt and others, and are (up to constants) best possible. Related to theabove sums, we also obtain upper and lower bounds for the cardinality of \[\{1\le n\le N:\|n\alpha-\gamma\|<\varepsilon\} \, , \] valid for allsufficiently large $N$ and all sufficiently small $\varepsilon$. This firststrand of the work is motivated by applications to multiplicative Diophantineapproximation, which are also considered. In particular, we obtain completeKhintchine type results for multiplicative simultaneous Diophantineapproximation on fibers in $\mathbb{R}^2$. The divergence result is the firstof its kind and represents an attempt of developing the concept of ubiquity tothe multiplicative setting.

Sums of reciprocals of fractional parts and multiplicative Diophantine approximation