Non-Archimedean Koksma Inequalities, Variation, and Fourier Analysis

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