SCHWEITZER 1994 Problem 4

For a given irrational number $\alpha$ , $y_{1,\alpha} = \alpha$ . If $y_{n-1, \alpha}$ is given, let $y_{n, \alpha}$ be the first member of the sequence $\big (\{k \alpha \} \big) ^ \infty_{k = 1}$ to fall in the interval $(0, y_{n-1,\alpha})$ ({ x } denotes the fraction of the number x ). Show that there exists an open set $G\subset (0,1)$ , which has a limit point 0 and for all irrational $\alpha$ , infinitely many members of the $(y_{n,\alpha})$ sequence do not belong to G.