Mahler coefficients of 1-Lipschitz measure-preserving functions on ℤp

Abstract

In this work, we provide a complete description of Mahler coefficients of [Formula: see text]-Lipschitz measure-preserving functions on the ring of [Formula: see text]-adic integers [Formula: see text]. Our techniques are mainly based on some congruence identities including binomial coefficients. The main result provides an answer to one of Jeong’s conjectures, concerning a characterization of [Formula: see text]-Lipschitz measure-preserving functions by means of their Mahler coefficients. We provide an example showing that the formula mentioned in Jeong’s conjecture is sufficient but not necessary. Namely, we prove that there exist [Formula: see text]-Lipschitz measure-preserving functions that do not satisfy Jeong’s formula.