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I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ satisfying this inequality, one has $$ N(x) = \frac{\log(2x)\log(3x)}{2\log2\log3} + o\left(\frac{\log x}{\log\log x} \right). $$ The error term here depends critically on the diophantine nature of the real number $\frac{\log 2}{\log 3}$, and in particular on the growth of the denominators $q_m$ of its convergents (in the sense of continued fractions). One could obtain a power savings in $\log x$ above if one knew something like $q_{m+1} \ll q_m^A$ for some real number $A$, but I suspect proving such a bound is very hard. My knowledge of diophantine approximation is introductory at best, so I would like to know from any experts the following:

  1. Are there "standard" conjectures which predict the growth of $q_m$, at least in the case when one is approximating a quotient of logarithms of integers?
  2. Hardy proves that $q_{m+1} \leq e^{A q_m}$ for some explicit constant $A$, which he improves (using an old theorem of Pillai) to $e^{\varepsilon q_m}$ for any $\varepsilon > 0$ with $m > m_0(\varepsilon)$ . I suspect not much more is known in terms of a better bound unconditionally. Are there improvements to this estimate?
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The keyword you are looking for is "irrationality measure" -- I think some authors (such as Lang) call it constant type. If you know the irrationality measure of $\alpha$ is $\mu = \mu(\alpha)$, then the convergents of $\alpha$ satisfy $q_{k+1} \ll q_k^{\mu -1 + \epsilon}$ for every $\epsilon>0$.

Most irrational numbers you can write down will have irrationality measure $2$. This can be made precise in the sense that a standard probabilistic argument shows that the set of $\alpha$ with $\mu(\alpha)>2$ has null Lebesgue measure. Showing this for explicit numbers, however, can be very hard -- for example, Roth's theorem (that famously probably won him a Fields medal) states that $\mu(\alpha) = 2$ if $\alpha$ is an algebraic irrational.

I'm not sure what is best known for the irrationality measures of quotients of logarithms, but I found this previous MO question when looking it up. Lucia seems to think that's the state of the art (at least circa 2014), and that gives something like $q_{k+1} \ll q_k^{7.617}$.

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  • $\begingroup$ Thanks for the answer. I knew about irrationality measure, but I didn't think to phrase it that way in terms of convergents. $\endgroup$ Dec 13, 2022 at 8:32
  • $\begingroup$ @JoshuaStucky: Sure! Do you mind sharing a reference for Hardy's result? It seems pretty cool. $\endgroup$ Dec 13, 2022 at 9:30
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    $\begingroup$ It's Chapter 5 of his book "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work" $\endgroup$ Dec 13, 2022 at 21:25
  • $\begingroup$ The original application I had in mind was to numbers $x < m^a n^b \leq c x$ with $m,n$ fixed and $a,b$ lying in prescribed residue classes, and $c$ constant or varying perhaps slightly with $x$. This requires a sufficiently good error term in the (generalized) counting problem I mentioned if you want to say something for all $x$ and not just sufficiently large $x$. $\endgroup$ Dec 13, 2022 at 21:30

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