# Prove $e^n$ and $\ln(n)$, mod 1, for $n=2,3,4…$ is dense in $[0,1]$

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 $0<a<b<1$, there is an integer $m$ such that $0<a<f(m) mod 1<b<1$

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They are not uniformly distributed, which would mean e.g. $$\lim_{n\to\infty} \frac{|\{k<n\colon f(n)<a\}|}n=a\quad\text{for all }a\in[0,1].$$ For example between $e^M$ and $e^{M+1}$ there are about $e^{M+1}-e^{M+\frac12}$ numbers $n$ with $\ln(n)\bmod 1>\frac 12$ and $e^{M+\frac 12}-e^{M}$ numbers $n$ with $\ln(n)\bmod 1<\frac 12$. These counts differ by a factor of $\sqrt e$ and that will be the relative proportion the larger the range of $n$ one checks becomes.

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 $N<n<e N+1$ hit every subinterval of length $\epsilon$.

For the exponential this is a bit more difficult.

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I do not know for $e^n$, but for $\log(n)$ modulo 1, it is not uniformly distributed, according to http://en.wikipedia.org/wiki/Equidistributed_sequence .