Non-Archimedean Koksma inequalities, variation, and Fourier analysis

TL;DR

This paper explores four notions of variation for functions on non-Archimedean local fields, establishing new inequalities and properties, with implications for regularity analysis and Fourier methods in non-Archimedean settings.

Contribution

It introduces a new, order-free, translation-invariant variation measure that admits a sharper Koksma inequality than previous versions.

Findings

The new variation is order-free and translation invariant.

The new Koksma inequality can be sharper than existing inequalities.

Different notions of variation have distinct regularity and inequality properties.

Abstract

We examine four different notions of variation for real-valued functions defined on the compact ring of integers of a non-Archimedean local field, with an emphasis on regularity properties of functions with finite variation, and on establishing non-Archimedean Koksma inequalities. The first version of variation is due to Taibleson, the second due to Beer, and the remaining two are new. Taibleson variation is the simplest of these, but it is a coarse measure of irregularity and it does not admit a Koksma inequality. Beer variation can be used to prove a Koksma inequality, but it is order-dependent and not translation invariant. We define a new version of variation which may be interpreted as the graph-theoretic variation when a function is naturally extended to a certain subtree of the Berkovich affine line. This variation is order-free and translation invariant, and it admits a Koksma inequality which, for a certain natural family of examples, is always sharper than Beer's. Finally, we define a Fourier-analytic variation and a corresponding Koksma inequality which is sometimes sharper than the Berkovich-analytic inequality.