A Remark of Schmidt's game

Schmidt's game is a powerful tool which was introduced by Schmidt. It is well known that a set which is winning in the sense of Schmidt's game has the full Hausdorff dimension and the countable stability. It has been proved that there exists an (α,β)-winning set which will not be winning if α decreases. In this paper, we shall construct a suitable fractal set S from expansions with integer base, and prove that S will not be winning if β increases.