cluster points of sub-sequences of sequence $\frac{n}{e}-[\frac{n}{e}]$

Question

what is the set of all cluster points of sub-sequences of sequence $\frac{n}{e}-[\frac{n}{e}]$ in which e is Euler's number 2.7182818284590452353602874713527...?

Answer 1

$\alpha:=1/e$ is irrational. Hence the sequence of fractional parts $ \left(\{n\alpha\}: n\ge 1\right) $ is equidistributed in $[0,1]$. So, the set of clusters (ordinary accumulation points) is still the whole interval $[0,1]$.

We can prove something stronger. Let $\Gamma_x$ be the set of statistical cluster points, i.e., the set of all $y$ such that $$ S_\varepsilon:=\{n: |x_n-y|<\varepsilon\} $$ has not asymptotic density zero for all $\varepsilon>0$, which means $\frac{|S_\varepsilon \cap [1,n]|}{n} \not\to 0$ as $n\to \infty$. (Clearly, this is a subset of the ordinary accumulation points.) However, in this case, it is still true that $\Gamma_x=[0,1]$.