On the digits of Schneider's p-adic continued fractions

Let Zp be the ring of p-adic integers, λ the Haar measure on pZp and an(x) the n-th digit of Schneider's p-adic continued fractions of x∈pZp. We prove that an(x) are independent and identically distributed with respect to λ. Moreover, we obtain the Hausdorff dimensions of some sets defined by the growth rate of an(x) and the sum of an(x) respectively. Such dimensional results are different from that in the cases of real numbers and formal series.