Onp-adic functions preserving Haar measure

Abstract

Let {a _n } ^=0/∞ _n be a uniformly distributed sequence ofp-adic integers. In the present paper we study continuous functions close to differentiable ones (with respect to thep-adic metric); for these functions, either the sequence {f(a _n )} ^=0/∞ _n is uniformly distributed over the ring ofp-adic integers or, for all sufficiently largek, the sequences {f _k (ϕ_k(a_n))} ^=0/∞ _n are uniformly distributed over the residue class ring modp ^k, whereϕ _k is the canonical epimorphism of the ring ofp-adic integers to the residue class ring modp ^k andf _k is the function induced byf on the residue class ring modp ^k (i.e.,f k _(x) =f(ϕ _k (x)) (modp ^k)). For instance, these functions can be used to construct generators of pseudorandom numbers.