I have a question that I can't manage to answer myself. It comes from some work in PDE theory, but it is related to analytic number theory.

Let us say that we have an irrational number $\xi$. The question is basically the following: is it true that the numbers $n \{n \xi\}$, where $\{\cdot\}$ stands for the fractional part, are dense in $\mathbb{R}^+$ as $n \in \mathbb{N}$?

~~If I make some numerical experiments, the answer seems to be affirmative~~. I know that $\{n\xi\}$ is uniformly distributed in $[0,1]$ and hence everywhere dense, but I need to know if *any* positive numer can be approximated by terms of the sequence $n \mapsto n \{n \xi\}$.

Any help or reference is really welcome.

**Edit**

As I said in the comments, I am interested in the behavior of the sequence $$ n \mapsto \left| 2\{n\xi\}-1 \right| n. \tag{1} $$ If I plot $$ n \mapsto \left( 2\{n\xi\}-1 \right)n,\tag2 $$ it seems to me that the sequence is uniformly distributed in $(-\infty,+\infty)$. Now (1) is just the absolute value of (2). Can we say anything about the absolute value of a u.d. sequence? I've found nothing in the most popular books.

allirrational $\xi$ or would almost all be enough? $\endgroup$ – Sean Eberhard Apr 16 '16 at 11:37Almost allwould be much more than I know ;-) $\endgroup$ – Siminore Apr 16 '16 at 11:58