How can one prove $e^n$ and $\ln(n)$, modulo 1, are dense in $[0,1]$, for $n=2,3,4...$?
By dense is meant, for any $0, there is an integer $m$ such that $0
How can one prove $e^n$ and $\ln(n)$, modulo 1, are dense in $[0,1]$, for $n=2,3,4...$?
By dense is meant, for any $0, there is an integer $m$ such that $0
They are not uniformly distributed, which would mean e.g. $\lim_{n\to\infty} \frac{|\{k
But they are dense in $[0,1]$ and that is the property you are looking for (as reflected by the edit of the question).
For the logarithm: Let $\epsilon>0$ be given. Find $N$ such that $\frac1N<\epsilon$. Then $0<\ln(n+1)-\ln n<\frac1n<\epsilon$ for all $n>N$ (because the derivative of $\ln$ is the reciprocal). Therefore the numbers $\ln n\bmod1$ with $N
For the exponential this is a bit more difficult.