"class" of dense subsets of the unit circle

Question

I have seen some questions on here about dense subsets of the unit circle, e.g. here or here. The proofs seem to always be somewhat similar (Pigeonhole principle), but there is always some special property of the given sequence that is being used. For me, this results in the following question: Given a sequence $(a_n)_n$ of real numbers with the property that \begin{equation} e^{ia_n}=e^{ia_m} \implies n=m, \end{equation} can we deduce that $\{e^{ia_n}:n \in \mathbb{N}\}$ is dense in the unit circle $\partial\mathbb{D}=\{|z|=1\}$?

Answer 1

To make sure this question is answered: No. As a counterexample, $\{e^{ia_n}; a_n = \frac{1}{n^2}$; $n=1,2,\ldots\}$ will certainly do.