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 $\log (p/q) \asymp H^{-1}$)

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 $\log (p/q) = \log(1 - 1/H) \asymp H^{-1}$)