Dimensions of Orbit Closures and Discrepancy for Dynamical $p$-adic Sequences

Classical discrepancy quantifies the irregularity of the distribution of a sequence in the unit interval. In this paper, we study the analogous notion for sequences in the ring of $p$-adic integers with a focus on the dynamically generated sequences. We prove that the orbits of ergodic $1$-Lipschitz self-maps of $\mathbb{Z}_p^d$ attain the optimal order of discrepancy and hence form low-discrepancy sequences. We also obtain bounds on the growth of the size of orbits of polynomial self-maps of $f: \mathbb{Z}_p^d \to \mathbb{Z}_p^d$ modulo $p^n$ for $d>1$. As a consequence, we show that orbit closures of $f$ have box dimension either zero or one. Our approach relies on the introduction of strong fixed points for such maps, together with several decomposition results for matrices over $\mathbb{Z}_p$.