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

Let Z p be the ring of p-adic integers, λ the Haar measure on p Z p and a n ( x ) the n-th digit of Schneider's p-adic continued fractions of x ∈ p Z p . We prove that a n ( 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 a n ( x ) and the sum of a n ( x ) respectively. Such dimensional results are different from that in the cases of real numbers and formal series.