Non-Archimedean Koksma Theorems and Dimensions of Exceptional Sets

TL;DR

This work develops a non-Archimedean analogue of Koksma's equidistribution theorem for local fields, proving that $([\,\alpha x^n\,])$ is uniformly distributed in the valuation ring for almost every $x$ when the characteristic is zero, and that the corresponding subsequences become uniform under a weighted measure in positive characteristic. A unifying metric framework based on expanding scaling maps yields these u.d. results and also explains large exceptional sets: for lacunary scaling maps, the exceptional sets have full Hausdorff dimension and possess rich $q$-homogeneous fractal structure. The paper further analyzes the dichotomy between Haar-u.d. and weighted-u.d. in the positive characteristic case, introduces $\,\\mu_k$ and $\\mu^*$ measures, and provides detailed metrical results and obstructions (e.g., Pisot-Chabauty phenomena) for non-uniform distribution in the non-Archimedean setting. Together, these results extend the understanding of uniform distribution and fractal exceptional sets in non-Archimedean dynamics and open pathways to p-adic analogues of real-analytic distribution problems.

Abstract

We establish a non-Archimedean analogue of Koksma's theorem. For a local field F of characteristic zero, we prove that the sequence ([αx^n]) is uniformly distributed in the valuation ring O for almost every x with |x|_p>1. In the case of positive characteristic, ([x^n]) fails to be uniformly distributed, but it becomes μ*-uniformly distributed for some weighted measure μ*. These results are derived from a general metric theorem for sequences generated by expanding scaling maps. On the other hand, we demonstrate that the exceptional set of parameters x for which these sequences are not uniformly distributed is large (i.e. having full Hausdorff dimension) and share a rich q-homogeneous fractal structure.

Figures (4)

• Figure 1: The field $\mathbb{Q}_p$ viewed as an infinite tree.
• Figure 2: A $3$-homogeneous set with $0,2,4\in I_{\Omega}$ .
• Figure :
• Figure :

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Lemma 3.1: $\tilde{f}$-invariance of Haar measure
• proof
• Lemma 3.2