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 $n$?
