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In his blog Terence Tao discusses the distance between powers of 2 and 3 and presents the following corollary:


Corollary 4 (Separation between powers of {2} and powers of {3})

  • For any positive integers {p, q} one has

    $$ \displaystyle |3^p - 2^q| \geq \frac{c}{q^C} 3^p$$

    for some effectively computable constants {c, C > 0} (which may be slightly different from those in Proposition 3).


What does he mean with "... effectively computable constants ... " ?

I've only a guess so far based on the inspection of the curve for $p$ and $q$ (=$N$ and $S$ in my usual notational style) using $p$ from the first hundred or so of the convergents of the continued fraction of $\log_2 (3)$ giving data for $p$ up to $1e175$ (only convergents where $2^q > 3^p$ are used).

From this I guess for instance $c=0.005$ and $C=1.01$. But those guesses might be much too crude.
I already presented an older guess in a MO-answer of mine but which seems even cruder.

So my question:

Q: How can one compute that constants?


pictures making my guess. Used only that cases where $2^S > 3^N > 2^{S-1}$ that means also from the original convergents of the continued fractions only each second one.

Image for the whole tested interval:
pic1

Detail for the smaller leading interval:
pic2

Detail for the smaller critical interval at $N \approx 1e166$:
pic3

Picture rotated to make comparision better visible. Note that the labeling of the axes are now no more correct, and the apparent numbers $N$ are scaled due to rotation (note: the logs of all values were rotated using $\cos(),\sin()$ by $45$ deg).
pic4

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  • $\begingroup$ I'm assuming... by reading the proof of this Proposition 3 (or, rather, of Baker’s theorem of which it is a corollary), that presumably is proven in not only a constructive way, but by describing algorithmically how to obtain these constants. $\endgroup$ – Clement C. Sep 6 '17 at 12:09
  • $\begingroup$ @ClementC.- yes, this might be. But in any case: I'd like to see such a constructive computation or algorithm. I've come across such phrase "is effectively computable" frequently in the last weeks, but never found such computation been described - perhaps I'm missing some basic understanding here. $\endgroup$ – Gottfried Helms Sep 6 '17 at 12:58
  • $\begingroup$ Well, have you had a look at the proof in question? $\endgroup$ – Clement C. Sep 6 '17 at 13:00
  • $\begingroup$ @ClementC.:Hmm, I've read that blog-entry many times (not only this week) and never found any thing what would show me how I could actually do that "effective computation". Perhaps there's some blind spot on my side... $\endgroup$ – Gottfried Helms Sep 6 '17 at 13:07
  • $\begingroup$ Oh, I am talking about the proof of Baker’s theorem... I don't know it, but as far as I can see it's not in that blog post. $\endgroup$ – Clement C. Sep 6 '17 at 13:08
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A method for computing the constants arising from linear forms of logarithms is clearly summarized here.

For the case where there are two logarithms, one can appeal to the results of Laurent, Mignotte, and Nesterenko for sharper constants (for references, see Exercise 4 from above, and see my related question).

Update 4/18/2018: One should also consider the work of Matveev (found in the references above) and G. Rhin ("Approximants de Padé et mesures effective s d'irrationalité").

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    $\begingroup$ Oh, thank you for the link. Unfortunately I haven't been able to go through this article of H-J- Evertsen a couple of years ago - I've got lost in space, so to say... Just recently I looked again at J. Simons papers on the disproof of the m-cycles in the Collatz-problem and took on to chew through the arguments around the result of G. Rhin - at least I seem to have understood this correctly (I showed my efforts in go.helms-net.de/math/collatz/Collatz_1cycledisproof.pdf for its open discussion of correctness) and so I think I have at least a glance of what is going on ... ;-) $\endgroup$ – Gottfried Helms Apr 18 '18 at 11:05
  • $\begingroup$ @GottfriedHelms Thanks for the references! $\endgroup$ – rukhin Apr 18 '18 at 18:20

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