This is a general question of ``locating the right field.'' Are there good sources and a fitting name of a research area (and, better yet, results) for the following problem class:
Given a finite set of real numbers $x_{1}, x_{2}, \ldots, x_{K}$, find their rational approximations $n_{1}/d, n_{2}/d, \ldots, n_{K}/d$, that have the same denominator $d$, which is as small as possible (for a given error tolerance).
I have looked at multi-dimensional continued fractions, but that research seems to pursue problems different from the above.