A recent development of the Davenport-Heilbronn method for diophantine inequalities is reexamined, and then applied to a class of problems in diophantine approximation. Among other things, an asymptotic formula is obtained for the number of solutions of the simultaneous inequalities $|n_j - \lambda_j n_0| <\varepsilon$ with square-free $n_j \in [1,N]$, whenever the positive real numbers $\lambda_1, \ldots, \lambda_r$ and $1$ are linearly independent over the rationals.
"Counting Diophantine Approximations." Funct. Approx. Comment. Math. 39 (2) 237 - 260, December 2008. https://doi.org/10.7169/facm/1229696574