Rademacher functions independent random Variables You call $r_n$=sign($sin^{2n}(\pi t))$ Rademacher functions. 

I read " the $r_n's$ are independant random variables"

Why is this true ? I don't get it...
Read the Wikipedia Rademacher system article. I think you want
$\;r_n = \textrm{sign}(\sin(2^n 2\pi t))\;$ instead. The  $r_n$ are orthogonal as functions on $\;t\in[0,1]\;$ and thus the random variables associated with them are independant.
because the probability of its values are independent such that $p ( r_1=e_1 , r_2=e_2 , ... , r_n=e_n ) = p ( r_1=e_1 )  p ( r_2=e_2 ) .... p ( r_n=e_n )$ where $p$ is the probability measure and $e$ are borel subsets of the space , if this satisfied then $r_n$'s are independent random variables , you can take some specific values of n it would be much easier to prove it , GOOD LUCK.