Modulus $p$ Vuza canons: generalities and resolution of the case $\left\lbrace 0,1,2^k\right\rbrace $ with $p =2$

TL;DR

This paper introduces a new approach to rhythmic Vuza canons using modulo p arithmetic, especially p=2, enabling rapid computation and providing a complete constructive solution for the case p=2, with potential for generalization.

Contribution

It presents the first complete constructive solution for modulo 2 Vuza canons, advancing the understanding of tilings and paving the way for broader applications.

Findings

Complete constructive solution for modulo 2 tiling

Rapid computation of Vuza canons using modulo p

Potential extension to general modulo p tilings

Abstract

Non-periodic tilings of $\mathbb{Z}_N$ are a difficult to obtain key to a wide range of complex mathematical issues. In a musician's eyes, they are called rhythmic Vuza canons. After a short summary of the main properties of rhythmic tiling canons and after stressing the importance of Vuza canons, this article presents a new approach: modulo $p$ Vuza canons. Working modulo $p$ (especially with $p=2$) allows Vuza canons to be computed very quickly. But going back to traditional Vuza canons requires understanding of when and how the tiling is a covering. The article's main result is a complete case construction of a modulo $2$ tiling. This first solution of a modulo $2$ tiling is proved in a constructive way, which, it is hoped, is expandable to every modulo $p$ tiling. Following this path may lead to a solution to the problem of building modulo $p$ tilings and thus a solution to the problem of easily finding traditional Vuza canons.