Subsequences

[Grigorios Kostakos]

Prove that the sequence $\alpha_n=\lfloor{\rm{e}}^n\rfloor\,,\; n\in\mathbb{N}$, where $\lfloor{\cdot}\rfloor$ is the floor function, has a subsequence with all its terms to being odd numbers and a subsequence with all its terms to being even numbers.


Note: I don't have a solution.

[Grigorios Kostakos]

It seems that here we have an open problem. See
Salem numbers and uniform distribution modulo 1