Let $(x_n)_{n \in \mathbf{N}}$ be a sequence of real numbers, we say that $(x_n)$ is uniformly distributed mod 1 (u.d. mod 1) if
$$\lim_{N \to \infty} \frac{|\{1\leq n \leq N : (x_n -\lfloor x_n \rfloor) \in (a,b) \}}{N} = b-a.$$
for any $a,b \in \mathbf{R}$ such that $0\leq a\leq b\leq 1$. Many easily defined sequences are u.d. mod 1; for instance, $(n^c)_{n \in \mathbf{N}}$ for any $c \in (0,1)$ (by Fejer's criterion). Obviously, $(n)_n$ is not u.d. mod 1. So the next natural questions would be:
Is $(\log{n})_{n \in \mathbf{N}}$ u.d. mod 1?
Surprisingly, (or at least to me at first) it is not. I found a proof in Kuipers & Niederreiter, pages 8-9, but I don't like it. I've tried to prove the result myself, using the direct definition and also Weyl's criterion for uniform distribution. But, I haven't been able to succeed. As someone once told me: "Any good theorem ought to be proved more than once". So, does anyone know of a different proof of this claim, or care to give me some hints on how to proceed?
Intuitively, this doesn't work because the spans of fractional parts of logs in an interval get longer and longer as we go up. To be more explicit, let us consider $a=0, b=0.1$ and I claim that the limit does not exist. For natural $k$, let $m=\lceil e^k \rceil$. Then let $m'=\lfloor e^{k+0.1} \rfloor$ For large enough $k$ the fraction in your limit is $\frac {m'-m+1}m \gt \exp(0.1-\epsilon) \gt 0.105$ because all the numbers $m$ through $m'$ have the fractional part between $0$ and $0.1$