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 $p = q1$, indeed in that case by a Taylor expansion $\log (p/q) = \log(1  1/H) \asymp H^{1}$)
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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 $\\log(p/q)\ = o(1/H^2)$. 

