Hausdorff dimension of the recurrence set of Gauss transformation

摘要

Let ${x_0in [0,1)}$ be an irrational number and {t n } n≥1 be a nondecreasing sequence of natural numbers. The recurrence set of Gauss transformation T is defined by $$E(x_0)={xin[0,1):T^n(x)in I_{t_n}(x_0) for infinitely many n},$$ where ${I_{t_n}(x_0)}$ denotes t n -th order cylinder of x 0 in continued fraction expansion. As a continuation of the work of Fernández, Melián and Pestana, we give the exact Hausdorff dimension of the set E(x 0).