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.
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 $\phi^n=dddd.999\dots999dddd$).