# Is nsinnn \sin n dense on the real line?

Is $\{n \sin n | n \in \mathbb{N}\}$ dense on the real line?

If so, is $\{n^p \sin n | n \in \mathbb{N}\}$ dense for all $p>0$?

This seems much harder than showing that $\sin n$ is dense on [-1,1], which is easy to show.

EDIT: This seems a bit harder than the following related problem, which might give some insight:

When is $\{n^p [ \sqrt{2} n ] | n \in \mathbb{N}\}$ dense on the real line, where $[\cdot]$ is the fractional part of the expression?

I am thinking that there should be some probabilistic argument for these things.

EDIT 2:

Ok, so plotting a histogram over $n \sin n$ is similar to plotting
$n \sin(2\pi X)$ where $X$ is a uniform distribution on $[-1,1].$
This is not surprising, since $n$ mod $2\pi$ is distributed uniformly on $[0,2\pi].$

Now, the pdf of $\sin(2\pi X)$ is given by $f(x)=\frac{2}{\pi \sqrt{1-x^2}}$
in $(-1,1)$ and 0 outside this set.

The pdf for $n \sin(2\pi X)$ is $g_n(x)=\sum_{k=1}^n \frac{1}{nk} f(x/k)$
so the limit density is what we get when $n \rightarrow \infty.$
(This integrates to 1 over the real line).

Now, it should be straightforward to show that for any interval $[a,b],$
$\int_a^b g_n(x) dx \rightarrow 0$ as $n \rightarrow \infty.$

Thus, the series $g_n$ is “too flat” to be able to accumulate positive probability anywhere. (The gaussian distribution on the other hand, has positive integral on every interval).

The answer to the first question will depend on the details of rational approximations of $\pi$, perhaps more than is actually known. A plot of $n \sin(n)$ shows some interesting apparent structure, probably due to some very good rational approximations.