There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer.

Recently I found that the same result is true if we replace $\alpha n$ by $\alpha n^2$ or any polinomial p such that $p(0)=0$.

Could this result be generalised to other functions? Particularly I'm curious about sequences $\alpha 2^n$ and $\alpha F_n$ where by $F_n$ I denote n-th Fibonacci number.

Does anyone know anything about it?