Metric theorem and Hausdorff dimension on recurrence rate of Laurent series

摘要

We show that the recurrence rates of Laurent series about continued fractions almost surely coincide with their pointwise dimensions of the Haar measure. Moreover, let $E_{lpha,eta}$ be the set of points with lower and upper recurrence rates $lpha$, $eta$ ($0leq lpha leq etaleq infty$), we prove that all the sets $E_{lpha,eta}$ are of full Hausdorff dimension. Then the recurrence sets $E_{lpha,eta}$ have constant multifractal spectra.