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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)
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  • $\begingroup$ 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). $\endgroup$ – hardmath Aug 21 at 22:56
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    $\begingroup$ @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. $\endgroup$ – Xander Henderson Aug 21 at 23:22
  • $\begingroup$ @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. $\endgroup$ – ComptonScattering Aug 22 at 0:35
  • $\begingroup$ @XanderHenderson thank you $\endgroup$ – ComptonScattering Aug 22 at 0:36
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    $\begingroup$ Try: math.tamu.edu/~doug.hensley/SimultaneousDiophantine.pdf $\endgroup$ – O. S. Dawg Aug 23 at 0:43
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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

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  • $\begingroup$ Thanks eddie, I will digest this and may well come back to ask further questions. $\endgroup$ – ComptonScattering Aug 22 at 20:16

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