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Subsequences

Real Analysis
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Grigorios Kostakos
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Subsequences

#1

Post by Grigorios Kostakos » Sat Jan 20, 2018 5:16 am

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
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Grigorios Kostakos
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Re: Subsequences

#2

Post by Grigorios Kostakos » Sun Jan 28, 2018 12:58 am

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