Are there any well known algorithms for finding good rational approximations to sets of real numbers?
Given just two real numbers, I can use continued fractions to find a rational approximation to their ratio, then use the numerator and denominator; for example a "good" rational approximation to { $\sqrt{2}$, $\sqrt{3}$ } is {9, 11} and {40, 49} is better.
However, with sets of three or more reals, there are some obvious special cases, but I can't see any obvious solution for the general case: in particular, pairwise approximations won't give a "good" approximation for the whole set (in general), i.e. there will be sets of smaller integers which are a better approximation.
Any reasonable definition of good will do - even defining what would count as a "reasonable" definition of good is interesting (in my opinion) even if a rather trivial question! A suitably cunning definition of "good" might even suggest an algorithm... :-)
I do think this question is more mathematical than computational, although perhaps a little elementary for this site, but please suggest where else I could ask it, if it is not appropriate here (and you can't point me at an answer).