A hybrid inequality of Erd\"os-Tur\'an-Koksma for digital sequences

TL;DR

This paper establishes a hybrid inequality for digital sequences combining Walsh and fb-adic functions, covering all cases of sequences based on digit vector addition, thus advancing the understanding of their distribution properties.

Contribution

It introduces a new hybrid inequality of Erd"os-Turán-Koksma type for sequences constructed via digit vector addition, unifying different function systems in a comprehensive framework.

Findings

Proves a hybrid inequality combining Walsh and fb-adic functions.

Shows the inequality applies to all digit vector addition-based sequences.

Provides insights into the distribution properties of digital sequences.

Abstract

For bases $\mathbf{b}=(b_1,..., b_s)$ of $s$ not necessarily distinct integers $b_i\ge 2$, we prove a version of the inequality of \etk \ for the hybrid function system composed of the Walsh functions in base $\bfb^{(1)}=(b_1,..., b_{s_1})$ and, as second component, the $\bfb^{(2)}$-adic functions, $\bfb^{(2)}=(b_{s_1+1},..., b_s)$, with $s=s_1+s_2$, $s_1$ and $s_2$ not both equal to 0. Further, we point out why this choice of a hybrid function system covers all possible cases of sequences that employ addition of digit vectors as their main construction principle.