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 $q<N$ for any $N$? (generalising the Golden ratio)
Best Answer
To the best of my knowledge, there does not exist an analog of the continued fraction algorithm to do this. However, as my colleague Xander Henderson pointed out, you can use a popular lattice basis reduction algorithm, known as the LLL algorithm, to calculate simultaneous Diophantine approximations. Here is a link to the original paper on LLL that got me started (see Proposition 1.39). link to paper