Rational approximation of multiple irrationals

Background

The best rational approximations $$p/q$$ to an irrational $$\alpha$$ are defined by the property $$\left|\alpha - \frac{p}{q}\right| < \left|\alpha - \frac{p'}{q'}\right|$$ for all $$q' \leq q$$. The approximants $$p/q$$ are found by simply truncating the continued fraction expansion.

The "most" irrational number is the Golden ratio $$\phi$$, which is defined by the property that for any given $$N$$, it has the most good approximations which satisfy $$q < N$$.

Furthermore, for (i) algebraic and (ii) almost all irrational numbers, they satisfy the bound $$\left|\alpha - \frac{p}{q}\right| > \frac{1}{q^{2+\epsilon}}$$ for any $$\epsilon > 0$$ and $$q$$ sufficiently large.

Context

I am interest in known generalisations of these results to the approximation of multiple irrationals.

I have found a generalisation of part of the final result, which is provided by the Subspace theorem. The subspace theorem has the following corollary: for $$D$$ rationally independent algebraic numbers $$(\alpha_1, \alpha_2, \ldots \alpha_D)$$, $$\left|\alpha_d - \frac{p_d}{q}\right| > \frac{1}{q^{1+1/D+\epsilon}}$$ for any $$\epsilon > 0$$, and $$q$$ sufficiently large.

Questions

My questions are:

• Is there a commonly used corresponding definition of the best rational approximations $$(p_1/q,p_2/q \ldots p_D/q)$$ to the irrational tuple $$(\alpha_1, \alpha_2, \ldots \alpha_D)$$? (generalising the first equation above)
• If there is a good definition, is there a better method than exhaustive search for finding the rational approximations $$p_d/q$$ to the irrational tuple $$\alpha_d$$? (generalising the truncated continued fraction expansion)
• For a given $$D$$ is there a known "most irrational" tuple $$(\alpha_1, \alpha_2, \ldots \alpha_D)$$ in the sense that there are the maximal number of good approximations satisfying $$q for any $$N$$? (generalising the Golden ratio)
• Although "truncations" of continued fractions are best rational approximations, there are some additional fractions related to the continued fraction expansion which are also best rational approximations. It's been awhile since I looked at the literature on simultaneous rational approximations, but I recall no analogy to the continued fraction algorithm (for multiple targets). – hardmath Aug 21 at 22:56
• @hardmath Indeed, my understanding is that there really aren't any good algorithms. I think that there is a way of finding good simultaneous approximations using a combination of continued fraction expansion (to get initial approximants), then massaging these a bit further with a lattice basis reduction algorithm such as LLL. I have a colleague who wrote his PhD thesis on this---I'll give him a ping and see if he has any insight. – Xander Henderson Aug 21 at 23:22
• @hardmath yes you are right, I should have written $|q \alpha < p| < | q' \alpha < p'|$ for the first equation and similarly multiplied by $q$ for all other equations. – ComptonScattering Aug 22 at 0:35
• @XanderHenderson thank you – ComptonScattering Aug 22 at 0:36
• – O. S. Dawg Aug 23 at 0:43