0
$\begingroup$

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

$\endgroup$
3
  • 1
    $\begingroup$ mathoverflow.net/questions/33257/… $\endgroup$ Jun 30, 2012 at 21:51
  • $\begingroup$ I also would like to know the conjectures regarding the "truth" $\endgroup$
    – kiskis
    Jun 30, 2012 at 22:18
  • $\begingroup$ As Mahler's conjecture suggest these could be much stronger than what follows from Baker $\endgroup$
    – kiskis
    Jun 30, 2012 at 22:18

1 Answer 1

4
$\begingroup$

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 $\|\log(p/q)\| = o(1/H^2)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.