Hausdorff dimension of the recurrence set of Gauss transformation

Abstract

Let \({x_0\in [0,1)}\) be an irrational number and {tn}n≥1 be a nondecreasing sequence of natural numbers. The recurrence set of Gauss transformation T is defined by $$E(x_0)=\{x\in[0,1):T^n(x)\in I_{t_n}(x_0)\ for\ infinitely\ many\ n\},$$ where \({I_{t_n}(x_0)}\) denotes tn-th order cylinder of x0 in continued fraction expansion. As a continuation of the work of Fernandez, Melian and Pestana, we give the exact Hausdorff dimension of the set E(x0).