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<p>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.</p>
</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.

Non-Archimedean Koksma Inequalities, Variation, and Fourier Analysis

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Non-Archimedean Koksma Inequalities, Variation, and Fourier Analysis"}

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...

Non-Archimedean Koksma Inequalities, Variation, and Fourier Analysis

Published Online: Dec 12, 2022

Page range: 21 - 50

Received: Mar 15, 2022

Accepted: Apr 17, 2022

DOI: https://doi.org/10.2478/udt-2022-0011

Keywords
-adic variation, equidistribution, Koksma inequalities, discrepancy, -adic Fourier analysis

© 2022 Clayton Petsche et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Non-Archimedean Koksma Inequalities, Variation, and Fourier Analysis

Published Online: Dec 12, 2022

Page range: 21 - 50

Received: Mar 15, 2022

Accepted: Apr 17, 2022

DOI: https://doi.org/10.2478/udt-2022-0011

Keywords-adic variation, equidistribution, Koksma inequalities, discrepancy, -adic Fourier analysis

© 2022 Clayton Petsche et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
-adic variation, equidistribution, Koksma inequalities, discrepancy, -adic Fourier analysis