\(1/2\)-heavy sequences driven by rotation (Q478502)

For \(x\in [0,1)\) let \(f(x)=\chi_{[0,1/2]}(x)-\chi_{(1/2,1)}(x)\), \(T(x)=x+\theta\) \(\mod 1\) for irrational \(\theta\), \(S_n(x)=\sum_{i=0}^{n-1}f\circ T^i (x)\), and \[ H_{\theta}=\{x\in [0,1): S_n(x)\geq 0, n=1,2,\ldots\}, \] \[ H_{\theta}^*=\{x\in [0,1): S_n(x)> 0, n=1,2,\ldots\}. \] The author shows that for every irrational \(\theta\), \(H_{\theta}^*\) is a singleton. As a corollary, it follows that \(H_{\theta}^*=\{0\}\) if and only if all partial quotients of \(\theta\) of odd index are themselves even. As for the Hausdorff dimension for \(H_{\theta}\), the author proves that there is some constant \(c\in (0,1)\) such that for almost-every \(\theta\), \(\dim_H(H_{\theta})=c\). Moreover, it is shown that given any \(d\in[0,1]\), there is a dense set of \(\theta\) for which \(\dim_{H}(H_{\theta})=d\).