Let $a,b\in\Bbb N$ be fixed. What are the best known lower bounds on $|a^n-b^m|$ for $n,m\in\Bbb N$, provided the difference is not $0$?
If $\frac{\log a}{\log b}$ is rational, then $a,b$ are of the form $c^k,c^l$ for some $c,k,l\in\Bbb N$, hence the difference between $a^n,b^m$ is of exponential size (indeed, I believe it's at least $\min\{a^n,b^m\}$).
The interesting and far less trivial are the bounds when $\frac{\log a}{\log b}$ is irrational, e.g. $a=2,b=3$. Using method sketched in this blog post we can prove that for some constants $c,C$ depending on $a,b$ we have $|a^n-b^m|>\frac{c}{m^C}a^n$.
Is this the best known estimate on $|a^n-b^m|$ known?
Thanks in advance.
In a certain sense, yes. Continued fractions provide an infinite number of pairs $m,n$ such that $\big|\frac{\log a}{\log b} - \frac{m}{n}\big| < \frac{1}{n^2}$. For such pairs we have $$\left|1 - b^m/a^n\right| = \left| 1 - \big(b^{\frac{m}{n} - \frac{\log a}{\log b}}\big)^n\right| = \left| 1 - b^{O(1/n)} \right| = O(1/n),$$ with constants depending on $a,b$. Thus we already have $|a^n - b^m| < \frac{c}{m} a^n$ for infinitely many values of $m,n$, which is of the same form as stated in your question. The main difference would seem to be in the value of the exponent $C$, which is related to the irrationality measure of $\log a/\log b$.