Is {nsinn|n∈N} dense on the real line?

If so, is {npsinn|n∈N} dense for all p>0?

This seems much harder than showing that sinn 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 {np[√2n]|n∈N} dense on the real line, where [⋅] 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 nsinn is similar to plotting

nsin(2πX) where X is a uniform distribution on [−1,1].

This is not surprising, since n mod 2π is distributed uniformly on [0,2π].Now, the pdf of sin(2πX) is given by f(x)=2π√1−x2

in (−1,1) and 0 outside this set.The pdf for nsin(2πX) is gn(x)=∑nk=11nkf(x/k)

so the limit density is what we get when n→∞.

(This integrates to 1 over the real line).Now, it should be straightforward to show that for any interval [a,b],

∫bagn(x)dx→0 as n→∞.Thus, the series gn is “too flat” to be able to accumulate positive probability anywhere. (The gaussian distribution on the other hand, has positive integral on every interval).

**Answer**

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

**Attribution***Source : Link , Question Author : Per Alexandersson , Answer Author : Robert Israel*