  There are two main interrelated goals of this paper. Firstly we investigate
the sums \[
S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}~~~\text{and}~~~
R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|}\,, \] where
$\alpha$ and $\gamma$ are real parameters and $\|\cdot\|$ is the distance 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 \[
\{1\le n\le N:\|n\alpha-\gamma\|<\varepsilon\} \, , \] 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.
