Ergodicity of integer multiplication on the circle
= Ergodicity of integer multiplication on the circle
The <integer multiplication map on the circle> is an <ergodic transformation> for <Lebesgue measure>. In the <Fourier basis>, an invariant function has coefficients $c_j=0$ when $K$ does not divide $j$, and $c_j=c_{j/K}$ otherwise. Every nonzero index eventually reduces to the first case, so only the constant coefficient remains.