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I need to prove that the sequence $\{\gamma_n\}$ where $\gamma_n$ is the fractional part of $\big(\frac{1+ \sqrt{5}}{2}\big)^n$ is NOT equidistributed in $[0,1]$.

Now, I am not sure if I am correct but if not then please correct me. A sequence in some interval is equidistributed if it is dense in that interval. Right?? So, I was thinking of proving that $\gamma_n$ is dense in $[0,1]$ but I do not have any clue how to start. I am looking for a hint/answer that is from Fourier analytic point of view rather than number theory.

user383517
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  • I was thinking of proving it using the concepts of Fourier analysis.. rather than from a number theoretic point of view! – user383517 Sep 29 '16 at 05:49
  • @user372236 Possible, since according to the book "Fractal Geometry and Stochastics II", by Brandt, Graf and Zhäle, A Cantor set with dissection ratio $\lambda$ is a set of uniqueness for Fourier series iff $\lambda^{-1}$ is a Pisot number. (quoting Salem). However, I can't help you much on this. Maybe there is a simple method, let's see if there are other answers. – Jean-Claude Arbaut Sep 29 '16 at 06:02
  • Ok.. I was also thinking of applying Weyl's criterion of equidistribution. – user383517 Sep 29 '16 at 06:05

1 Answers1

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Hint:

Have a look at Pisot numbers on Wikipedia.

Then prove that $\phi=\dfrac{1+\sqrt5}{2}$ is a Pisot number. You may use the fact that it's a root of the polynomial $x^2-x-1$, and that the other root has absolute value $<1$. You may also use Newton's identities.

Concretely, you should be able to prove that $\phi^n$ is nearer and nearer to an integer as $n$ grows. Hence its values $\bmod 1$ have two limit points: $0$ (for $\phi^n=dddd.000\dots000dddd$) and $1$ (for $\phi^n=dddd.999\dots999dddd$).

Jean-Claude Arbaut
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  • I was just about to add the suggestion that the OP should do some testing and calculate fractional parts of powers of $\phi^n$... :-) IOW, I highly approve of this hint. – Jyrki Lahtonen Sep 29 '16 at 05:46
  • Can we prove using the concepts of Fourier analysis.. rather than from a number theoretic point of view! – user383517 Sep 29 '16 at 05:50