For the ring of $p$-adic integers, $p$ being a fixed prime, any sequence which plays a similar role to Weyl's irrational rotation has not been proposed yet. We will see that a modified $p$-adic van der Corput sequence provides us with a reasonable counterpart of Weyl's irrational rotation in the ring. We will present a similar random Weyl sampling on the ring to the one proposed by Sugita and Takanobu. In the process of establishing the counterpart, a sampling method based on a function with naturally extended domain to the field of $p$-adic numbers in terms of the additive characters will be mentioned.
"A pairwise independent random sampling method in the ring of $p$-adic integers." Osaka J. Math. 53 (3) 775 - 798, July 2016.