Get stuck when approaching this problem. Thanks for any suggestions in advance.
Given two positive irrationals $a$ and $b$. Denote $f(x)=x-[\frac{x}{a}]a$ (brackets mean the floor function). Proof that for any given positive integer $x_0$, there exists another positive integer $x_1$ s.t. $f(x_0+b)>f(x_1+b)$.
Furthermore, given such $x_0$, is it possible to find out the minimum $x_1$ satisfying the above constraint?