TITLE: Fractional part of rational power arbitrary small
QUESTION [7 upvotes]: I think that $\{a^n\}$ (where $\{x\}$ is $x \pmod 1$), where $a$ is fixed rational greater than 1 and $n$ is positive integer, is dense in $[0,1]$ is unsolved. However what about $\{a^n\}$ is arbitrary small for some $n$ ($a$ is fixed rational as well).

REPLY [7 votes]: I think that the wolfram link cited the first result (Vijayaraghavan 1941) incorrectly where it reports that 

$\{ (3/2)^n \}$ has infinitely many accumulation points in both $[0, 1/2)$ and $[1/2, 1]$.

The first result actually is 

$\{ (3/2)^n \}$ has infinitely many accumulation points in $[0,1]$. 

The second result cited by wolfram is by Flatto, Lagarias, Pollington (1995) which is available here: http://matwbn.icm.edu.pl/ksiazki/aa/aa70/aa7023.pdf. In the introduction of this paper by them, it reports that

Vijayaraghavan later remarked that it was striking that
  one could not even decide whether or not $(3/2)^n$ mod $1$ has infinitely
  many limit points in $[0, 1/2)$ or in $[1/2, 1)$.

This paper proved that 

$$\limsup\{ (3/2)^n \} - \liminf \{(3/2)^n\} \geq \frac13. $$

A few later results are mentioned (Dubickas 2006, 2008) in Yann Bugeaud's book "Distribution Modulo One and Diophantine Approximation" (for example, p. xi, p. 67 and p. 68). 

$\{ (3/2)^n \}$ has a limit point in $[0.238117 . . . , 0.761882 . . .]$ which has length $0.523764 . . .$.
$\{(3/2)^n\}$ has a limit point in $[0, 8/39] \cup
[18/39, 21/39] \cup [31/39, 1]$, of total length $19/39$.