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Denote by $||x||$ the distance between $x$ and the nearest integer. Mahler conjectured that there is a constant $c > 0$ such that for any integer $n \geq 2$ $$ ||\log n|| \geq n^{-c} $$ and Waldschmidt made the stronger conjecture that $$ ||\log n|| \geq (\log n)^{-c} $$ I am interested in the analogous question for rational numbers rather than integers. Namely, given two integers $0 < p \neq q \leq H$ are there any results and/or conjectures regarding how small $$ ||\log (p / q)|| $$ can be in terms of $H$ ? A lower bound of the form $H^{-1 - \varepsilon}$, for some small $\varepsilon > 0$ would be ideal but I'm not sure how much one can hope for. (Notice that $H^{-1}$ is attained when $q = H$ and $p = q-1$, indeed in that case by a Taylor expansion $\log (p/q) = \log(1 - 1/H) \asymp H^{-1}$)

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I also would like to know the conjectures regarding the "truth" – kiskis Jun 30 '12 at 22:18
As Mahler's conjecture suggest these could be much stronger than what follows from Baker – kiskis Jun 30 '12 at 22:18

Since the simple continued fraction for $e$ is known, and has arbitrarily large elements, for any $\epsilon > 0$ there are $p>q$ with $|p/q - e| < \epsilon/p^2$, and thus $\|\log(p/q)\| = o(1/H^2)$.

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