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Suppose $\alpha >1$ such that the distance from $\alpha^n$ to the integers tends to zero as $n$ tends to infinity.

Question 1: Can $\alpha$ be a rational number?

Question 2: The property is satisfied if $\alpha$ is a root of a monic polynomial with integer coefficient that has all the other roots with module less than 1. Is the converse true?

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1) No, it can not unless it is integer. 2) This is an open problem known as Pisot-Vijayaraghavan problem. For algebraic numbers the answer is positive, that implies the answer to 1). See https://en.m.wikipedia.org/wiki/Pisot-Vijayaraghavan_number

Short self-contained answer to 1): if $\alpha^n=A_n+\delta_n$ for positive integer $A_n$ and $\delta_n\to 0$, and $\alpha=p/q$, we get $qA_{n+1}-pA_n=-q\delta_{n+1}+p\delta_n\to 0$, but $qA_{n+1}-pA_n$ is integer, hence for large enough $n$ we have $A_{n+1}=\frac pq A_n$, this is impossible since such a sequence takes non-integer values for coprime $p,q>1$.

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    $\begingroup$ Could you give me a hint for 1) ? I have tried to prove it but failed. $\endgroup$
    – Pluviophile
    Commented Aug 13, 2016 at 8:01
  • $\begingroup$ Added to the answer. $\endgroup$
    – Fedor Petrov
    Commented Aug 13, 2016 at 9:59
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    $\begingroup$ In particular, I believe it's an open problem whether the distance from $e^n$ to the nearest integer tends to $0$. $\endgroup$
    – Greg Martin
    Commented Aug 13, 2016 at 16:36

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