Non-Archimedean Koksma Theorems and Dimensions of Exceptional Sets

TL;DR

This paper extends Koksma's theorem to non-Archimedean local fields, showing uniform distribution properties of certain sequences and analyzing the size and structure of exceptional sets where these properties fail.

Contribution

It establishes a non-Archimedean analogue of Koksma's theorem and characterizes the fractal structure of exceptional parameter sets.

Findings

Sequences are uniformly distributed for almost every x with |x|_p>1 in characteristic zero.

In positive characteristic, sequences are {}-uniformly distributed under a weighted measure.

Exceptional sets where uniform distribution fails have full Hausdorff dimension and fractal structure.

Abstract

We establish a non-Archimedean analogue of Koksma's theorem. For a local field F of characteristic zero, we prove that the sequence ([{\alpha}x^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 {\mu}*-uniformly distributed for some weighted measure {\mu}*. 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.