AI 中文总结
该研究针对p进整数环中的动力序列，证明遍历1-Lipschitz自映射的轨道为低偏差序列，得到多项式自映射轨道大小增长界，确定其轨道闭包盒维数为0或1。
AI 中文摘要
经典偏差量化了序列在单位区间中分布的不规则性。本文针对p进整数环中的序列研究类似概念，重点关注动力生成的序列。我们证明了Zₚᵈ上遍历1-Lipschitz自映射的轨道达到偏差的最优阶，因此构成低偏差序列。我们还得到了f: Zₚᵈ→Zₚᵈ（d>1）的多项式自映射模pⁿ的轨道大小增长的界，由此证明f的轨道闭包的盒维数为0或1。我们的方法依赖于为这类映射引入强不动点，以及Zₚ上矩阵的若干分解结果。
英文摘要
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 $\mathbb{Z}_p$.
CommentsA new case distinction between p=2 and odd primes
