Multiple tilings associated to d-Bonacci beta-expansions

TL;DR

This paper characterizes when Rauzy fractals from symmetric $eta$-expansions form tilings, especially for $d$-Bonacci numbers, where they create multiple tilings with a specific covering degree.

Contribution

It provides a necessary and sufficient condition for Rauzy fractals to form tilings and demonstrates multiple tilings for $d$-Bonacci numbers with a precise covering degree.

Findings

Rauzy fractals form tilings under certain conditions.

For $d$-Bonacci numbers, Rauzy fractals create multiple tilings.

The covering degree for these multiple tilings is $d-1$.

Abstract

Let $\beta\in(1,2)$ be a Pisot unit and consider the symmetric $\beta$-expansions. We give a necessary and sufficient condition for the associated Rauzy fractals to form a tiling of the contractive hyperplane. For $\beta$ a $d$-Bonacci number, i.e., Pisot root of $x^d-x^{d-1}-\dots-x-1$ we show that the Rauzy fractals form a multiple tiling with covering degree $d-1$.