On \(p\)-adic functions preserving Haar measure (Q1280619)

Let \(f(x)\) be a function from the set \(\mathbb{I}_p\) of all \(p\)-adic integers into itself, and \(\varphi_k: \mathbb{I}_p\to \mathbb{Z}/p^k\) be the canonical epimorphism of the ring \(\mathbb{I}_p\) onto the residue class ring \(\mathbb{Z}/p^k\) module \(p^k\), and \(f_k(x)\) be the function induced on \(\mathbb{Z}/p^k\) by \(f(x)\) (i.e., \(f_k (x)= f(\varphi_k(x)) \pmod{p^k})\). Suppose that \(\{a_n\}^\infty_{n=1}\) is a uniformly distributed sequence of \(p\)-adic integers. The author studies continuous functions close to differentiable ones (with respect to the \(p\)-adic metric) such that either the sequence \(\{f(a_n)\}^\infty_{n=0}\) is uniformly distributed over \(\mathbb{I}_p\) or the sequences \(\{f_k(\varphi_k (a_n))\}^\infty_{n= 0}\) are uniformly distributed over \(\mathbb{Z}/p^k\) for all sufficiently large \(k\). This work is of a purely mathematical nature, but it can be used to construct generators of pseudorandom numbers. The subject and the method in this paper are very close to the paper by \textit{V. S. Anashin} [Mat. Zametki, 55, No. 2, 3-46 (1994; Zbl 0835.11031)].