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It is an open problem whether the fractional part of $\left(\dfrac32\right)^n$ is dense in $[0...1]$.

The problem is: is $1$ a limit point of the above sequence?

An equivalent formulation is: $\forall \epsilon > 0: \exists n \in \Bbb N: 1 - \{1.5^n\} < \epsilon$ where $\{x\}$ denotes the fractional part of $x$.

Here is a table of $n$ against $\epsilon$ that I computed:

$\begin{array}{|c|c|}\hline \epsilon & n \\\hline 1 & 1 \\\hline 0.5 & 5 \\\hline 0.4 & 8 \\\hline 0.35 & 10 \\\hline 0.3 & 12 \\\hline 0.1 & 14 \\\hline 0.05 & 46 \\\hline 0.01 & 157 \\\hline 0.005 & 163 \\\hline 0.001 & 1256 \\\hline 0.0005 & 2677 \\\hline 0.0001 & 8093 \\\hline 0.00001 & 49304 \\\hline 0.000005 & 158643 \\\hline 0.0000005 & 835999 \\\hline \end{array}$

References

  1. Unsolved Problems, edited by O. Strauch, in section 2.4 Exponential sequences it is explicitly mentioned that both questions whether $(3/2)^n\bmod 1$ is dense in $[0,1]$ and whether it is uniformly distributed in $[0,1]$ are open conjectures.
  2. Power Fractional Parts, on Wolfram Mathworld, "just because the Internet says so"
DHMO
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  • If you are interested in either $0$ or $1$, you might be able to study this by thinking of it as a circle map. I don't know much about the dynamics of circle maps, but I know they are studied in dynamic systems, and much is known about them. – A. Thomas Yerger Feb 04 '17 at 08:08
  • This question is - in some sense - dual. It basically asks whether $0$ is a limit point. (And the other difference is that it asks about arbitrary rational number, not just $3/2$.) – Martin Sleziak Feb 04 '17 at 08:15
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    @AlfredYerger it's not a dynamical system because ${ \frac 32 x}$ isn't a function of ${x }$ – mercio Feb 04 '17 at 08:48
  • I think this question may very well be equivalent to the original open problem that the set of $\left{\left(\frac{3}{2}\right)^n\right}$ for $n\in\mathbb{N}$ is dense in $[0,1]$. – Batominovski Feb 04 '17 at 13:07
  • @Batominovski Can you explain why? – DHMO Feb 04 '17 at 13:08
  • It is a hunch. I can be very wrong. – Batominovski Feb 04 '17 at 13:10
  • I thought so originally too because I was thinking that ${x^2} = {x}^2$, but that's not true. – mercio Feb 04 '17 at 16:13
  • May I ask, with what sort of routine you've found that large $n=835999$? A brute search seems out of reason... – Gottfried Helms Feb 05 '17 at 06:44
  • @GottfriedHelms Indeed, a brute search. – DHMO Feb 05 '17 at 06:46
  • So ... 50 point bounty for an open problem? – zhw. Feb 13 '17 at 23:22
  • @zhw If you can find some source saying that this is open problem (or show that this is equivalent to some of known open problems about the sequences $(3/2)^n$), I'd consider that as a satisfactory answer. (At least satisfactory enough to be rewarded by a bounty.) – Martin Sleziak Feb 15 '17 at 06:01
  • I thought the reference 1. said "unsolved problem". I see now that it was two stonger conclusions that are open. – zhw. Feb 15 '17 at 21:55

1 Answers1

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Another comment, but too big for the standard box. An atanh() rescaling might be an interesting thing, see my example:

picture

The pink and the blue lines are hullcurves connecting the points $\small (N,f(N))$ where $f(N)$ is extremal (with moving maxima/minima) and the grey dots are points $\small (N,f(N))$ at $\small N \le 1000 $ which shall illustrate the general random distribution of the $\small f(N)$.
The grey lines are manually taken smooth subsets of the extremaldata and symmetrized (by merging of datasets and adapting sign) to show the rough tendency of extension of the vertical intervals.

I liked it that the atanh()-scaling seem to suggest some roughly linear increase/decrease of the hullcurves.

[update] The data for the picture were extended by data from the OP and OEIS A153663 (magenta upper curve) and from the OEIS A081464 (blue lower curve). Note, that the OEIS has even more datapoints, but that needed excessive memory/time to compute the high powers of (3/2) and its fractional parts.

Gottfried Helms
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