Let $m,n$ be natural numbers such that $2^n - 1 \mid 3^m - 1$. By results from Bugeaud-Corvaja-Zannier, say Theorem 3 of this paper , we know that for any constant $C > 0$ we must have $m > Cn$ for sufficiently large $n$. Is it known whether we can do better than that? Heuristically I would expect $m$ to be very large compared to $n$.
I would be happy to see any improvement over $Cn$, or even an average statement that $m$ has to be large compared to $n$ most of the time.