About the sequence $a_n=\{\pi^n\}$

Question

Is the sequence $\{\pi^n\}=\pi^n-\lfloor\pi^n\rfloor$ dense? In other words for any given $\varepsilon>0$ and $t\in[0,1]$ is there a proper $n\in\mathbb{N}$ satisfying $|\{\pi^n\}-t|<\varepsilon$ ?

(*) What is the condition on $q$ to make the sequence $\{q^n\}$ dense?

I know the necessary and sufficient condition for $\{nq\}$ is $q\not\in\mathbb{Q}$.

Also, to make the question * nicer, extend it for all (positive and negative) integers.

@Peter he knows about $nq$ and asks about $q^n$ – Hagen von Eitzen May 01 '18 at 07:00 — Hagen von Eitzen, May 01 '18 at 07:00
@HagenvonEitzen You are right. I have not read carefully. – Peter May 01 '18 at 07:01 — Peter, May 01 '18 at 07:01

Robert Z · Answer 1 · 2018-05-01T07:21:16.463

5

For $(\ast)$ see Power Fractional Parts and the references there. Hardy and Littlewood (1914) proved that the sequence of fractional parts $q^n-\lfloor q^n\rfloor$ is equidistributed for almost all real numbers $q>1$. One exceptional number is the golden ratio. I don't know if $\pi$ is another exception, probably not. I did not find a specific result about $\pi$. I think that it is an open problem.

edited May 01 '18 at 07:21

answered May 01 '18 at 07:00

Robert Z

145,942

well, and what about (specifically) $\pi$? The statement you cite shows that it is very likely for $\pi$ to have that property, but is it known to be true in this specific case? – Thomas May 01 '18 at 07:07
@Thomas I am afraid that there is a reason why the MW reference makes no mention of $\pi$ provably having that property – Hagen von Eitzen May 01 '18 at 07:10
I am looking thorough the references, but I did not find a specific result about $\pi$. – Robert Z May 01 '18 at 07:13
ah, an open problem then ;-) – Thomas May 01 '18 at 07:13

About the sequence $a_n=\{\pi^n\}$

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