  The Diophantine sums $\sum_{n=1}^N \| n \alpha \|^{-1}$ and $\sum_{n=1}^N
n^{-1} \| n \alpha \|^{-1}$ appear in many different areas including the
ergodic theory of circle rotations, lattice point counting and random walks,
often in connection with Fourier analytic methods. Beresnevich, Haynes and
Velani gave estimates for these and related sums in terms of the Diophantine
approximation properties of $\alpha$ that are sharp up to a constant factor. In
the present paper, we remove the constant factor gap between the upper and the
lower estimates, and thus find the precise asymptotics for a wide class of
irrationals. Our methods apply to sums with the fractional part instead of the
distance from the nearest integer function, and to sums involving shifts $\| n
\alpha + \beta \|$ as well. We also comment on a higher dimensional
generalization of these sums.
