For what values of $x \in \Bbb R$ is the set $\{\cos(nx)\}_{n\in \Bbb N}$ dense in $[-1,1]$?

Question

I'm struggling with this problem, I think it may depend on the rationality or irrationality of x but I'm not sure. I also thought it would be useful to study the density of $e^{inx}$ in $S^1$ but I haven't been able to get anywhere.

Can you think of a set of values for $x$ that you can immediately rule out? This might give you a hint as to how to proceed. — legionwhale, Oct 22 '23 at 19:52
I only manage to discard the obvious ones, like 0, $\pi$, $2\pi$... — Jjvvrr, Oct 22 '23 at 20:03

score 0 · Answer 1 · answered Oct 22 '23 at 19:52

0

Hint: It is a good idea to consider the function $\phi: \Bbb R \to S^1$ given by $\phi(x) = e^{inx}$. If the collection $\mathcal M = \{e^{inx}\}_{n \in \Bbb N}$ is dense in $S^1$, then its projection onto the real line, namely $\operatorname{proj} \mathcal M = \{\cos nx\}_{n \in \Bbb N}$ is dense in $\operatorname{proj} S^1 = [-1,1]$.

answered Oct 22 '23 at 19:52

stoic-santiago

14,877

1

That was the idea I spent the most time on... but I can't get anything, could you give me a second clue? hahaha I'm really stuck on this – Jjvvrr Oct 22 '23 at 20:11

For what values ​of $x \in \Bbb R$ is the set $\{\cos(nx)\}_{n\in \Bbb N}$ dense in $[-1,1]$?

1 Answers1

For what values of $x \in \Bbb R$ is the set $\{\cos(nx)\}_{n\in \Bbb N}$ dense in $[-1,1]$?