Some new games and badly approximable numbers (Q795863)

W. M. Schmidt has defined the (\(\alpha\),\(\beta)\)-game and has applied it to the set of badly approximable numbers. In the present paper several variations of Schmidt's game are investigated. Let \(\epsilon\) '\(\in(0,1/2]\) and \(f_ 1,f_ 2,..\). be real functions on an interval I. Suppose that for each \(f_ n\) and each \(x\in I\) there is some open neighborhood \(O_ x\) of x and some \(\epsilon ''>0\) such that for any intervals \(J_ 1\) and \(J_ 2\) with \(x\in J_ 2\subseteq J_ 1\subseteq O_ x, \ell(J_ 2)/\ell(J_ 1)=\epsilon '\quad \Rightarrow \quad \epsilon ''\leq \ell(f_ n(J_ 2))/\ell(f_ n(J_ 1))\leq 1/2\)(\(\ell\) denotes the length of an interval). Then there exist in continuum many \(\alpha\) such that each of \(f_ 1(\alpha),f_ 2(\alpha),..\). is badly approximable.