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Is there any progress on a “baby $abc$ conjecture” where you restrict attention to rational approximations of $n$-th roots?

Let $r/s$ be a very close approximation to $(t/u)^{1/n}$, so that $$ |u\cdot r^n - t\cdot s^n| = a $$ is small. We thus have a pseudo-$abc$-triple with $c$ being the larger of $u\cdot r^n$ and $t\cdot s^n$, and $q=\frac{\log c}{\log(arstu)}$.

Example: the fifth root of $109$ is very close to $23/9$, so we get $$ 2+109\cdot 9^5=23^5,\quad 5\log(23)/\log(45126) = 1.4628, $$ not quite as good as the $1.6299$ it gets as an $abc$ triple where you use that $9 = 3^2$ but still good enough to make it the “best” known approximation of any root.

Is anything known about $q$ in terms of $n$?

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    $\begingroup$ Please use TeX on this site, and also use a high-level tag like "nt.number-theory". I added this tag now. $\endgroup$
    – GH from MO
    Jan 27, 2022 at 7:06
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    $\begingroup$ Applying Liouville's theorem on diophantine approximation to the algebraic number $(t/u)^{1/n}$ would get some preliminary bound. Improving this bound seems related to the problem of trying to find effective versions of the Thue-Siegel-Roth theorem. In this regard, the answers here may be relevant: mathoverflow.net/questions/58856/… $\endgroup$
    – Terry Tao
    Jan 27, 2022 at 16:16
  • $\begingroup$ I believe this approach is known for searching for high quality triples. $\endgroup$
    – joro
    Jan 27, 2022 at 17:49
  • $\begingroup$ The first page of Baker’s paper “Rational Approximations to Certain Algebraic Numbers” promises a constructive bound on the closeness of rational approximations of $nth$ roots of rationals, but the rest of the paper is behind a paywall and I can’t find even a non-paywalled STATEMENT of the theorem elsewhere! Can anyone give me a link to a statement of Baker’s result, or just state it here? $\endgroup$ Jan 28, 2022 at 1:07

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