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As part of my Phd thesis on aperiodic Wang tilings, I've discovered I need a bound on the irrationality measure of $\gamma = \log 2/\log 6$. That is, I am looking for an upper bound on the quantity

$\eta = \inf \{\alpha : \left|\frac{\log 2}{\log 6} - \frac{p}{q}\right| < \frac{1}{q^\alpha} \text{ for only finitely many $p,q\in\mathbf Z$}\}$.

I have found a paper An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers by E.M. Matveev. This paper gives a complicated system of parameters which can be adjusted bound the irrationality measure of rational linear combinations of logs. From this I have managed to obtain the bound

$\eta < 10^9$.

Does anyone know of references that can produce a better bound on this quanity? Perhaps one closer to its likely value of 2?

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See Georges Rhin: Approximations de Padé et mesures effectives d'irrationalité. (French) [Padé approximants and effective measures of irrationality] Séminaire de Théorie des Nombres, Paris 1985–86, 155–164, Progr. Math., 71, Birkhäuser Boston, Boston, MA, 1987.

Inequality (8) there shows that if $u_0$, $u_1$ and $u_2$ are integers with $H= \max(|u_1|,|u_2|)$ sufficiently large, then we have $$ |u_0+u_1\log 2 + u_2 \log 3| \ge H^{-7.616}. $$ Thus $\eta \le 8.616$.

I don't know a free online version for Rhin's paper (you can find it on Springer Link). But you can look at this paper by Qiang Wu in Math. Comp. which discusses similar problems and mentions Rhin's work (which seems the best known result for the $\log 2$ and $\log 3$ case).

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    $\begingroup$ This is great. I've contacted Rhin about his paper. Thank you. $\endgroup$
    – Jason Siefken
    May 15, 2014 at 0:22
  • $\begingroup$ Excuse me Lucia but could you explain the : "Thus η≤8.616" ? Thank you in advance. SXBnano $\endgroup$
    – SXBnano
    Dec 17, 2022 at 12:56
  • $\begingroup$ @SXBnano If log(2)/log(6) - p/q = +- 1/q^alpha, then q log(2) - p log(6) = +- 1/q^{alpha - 1}. Rewriting the LHS as (q - p) log(2) - p log(3), the theorem above asserts that 1/q^{alpha -1} >= max(p, q-p)^{-7.616}. As log(2)/log(6) is around 0.38 the q-p term is larger, and is at least q/2. So one finds 1/q^{alpha - 1} >= 2^{7.616} q^{-7.616}. Taking logarithms, (1-alpha) log(q) >= 7.6 log(2) - 7.616 log(q). Taking q large establishes alpha <= 8.616. $\endgroup$
    – mme
    Dec 17, 2022 at 14:48

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