Hausdorff dimension of the recurrence sets of Gauss transformation on the field of Laurent series (Q617793)

Let \({\mathbb F}_q\) be the finite field with \(q\) elements, \(I\) the set of all formal series \(\sum_{n=1}^\infty c_n z^{-n},c_n\in{\mathbb F}_q\), \(x_0\in I\) with the continued fraction expansion \([A_1(x_0),A_2(x_0),\ldots]\), \(t_n\) a non decreasing sequence of natural numbers, \[ \alpha:=\liminf_{n\to\infty}\frac{2\sum_{k=1}^{t_n}\deg A_k(x_0)}{n}, \] \[ E(x_0)=\{x\in I:T^n(x)\in I_{t_n}(x_0)\,\text{for infinitely many}\, n\}, \] where \(T\) is the Gauss transformation and \(I_{t_n}(x_0)\) the \(t_n\)-th order cylinder of \(x_0\). The authors prove that if \(1<\alpha<+\infty\) then the Hausdorff dimension of \(E(x_0)\) is \(s_\alpha\),the unique solution of \[ \sum_{k=1}^\infty (q-1)q^k\left(\frac{1}{q^{2k+\alpha}}\right)^s=1. \]