On the metric theory of p-adic continued fractions

An analogue of the regular continued fraction expansion for the p-adic numbers for prime p was given by T. Schneider, such that for x in pZp, i.e. the open unit ball in the p-adic numbers, we have uniquely determined sequences (bn∈{1,2,…,p−1},an∈N)(n=1,2,…) such that x=pa0b1+pa1b2+pa2b3+pa3b4+⋱. A sample result that we prove is that if pn(n=1,2,…) denotes the sequences of rational primes, we have limN→∞1N∑n=1Napn(x)=pp−1, almost everywhere with respect to Haar measure. In the case where pn is replaced by n this result is due to J. Hirsh and L. C. Washington. The proofs rely on pointwise subsequence and moving average ergodic theorems.