For $\epsilon > 0$, is there always $n,m \in \mathbb{N}$ so that $e^{n}$ is $\epsilon$-close to $m$?

Question

I don't have much to go off of, so I can't demonstrate any attempts here. I just want to know if there has been any answer or partial answer to this question.

Answer 1

You can make $e^a$ exactly equal to any positive number $y$ (integer or not) by setting $a = \ln y$. You can get as close as you like with rational $a$.

I suspect that $e^n$ comes arbitrarily close to an integer for integral $n$, but have not been able to find a reference. Weyl's theorem implies that the integral multiples of $e$ (or any other irrational number) are equidistributed modulo $1$.

The powers of $e$ probably are, but caution is called for. There are irrational numbers $\gamma$ whose powers are not equidistributed. (https://math.unm.edu/~crisp/courses/wavelets/fall13/wavelet-weyl-report2.pdf)