De Bruijn graphs and powers of $3/2$

Abstract

In this paper we consider the set ${\mathbb Z}^{\pm\omega}_{6}$ of two-way infinite words $\xi$ over the alphabet $\{0,1,2,3,4,5\}$ with the integer left part $\lfloor\xi\rfloor$ and the fractional right part $\{\xi\}$ separated by a radix point. For such words, the operation of multiplication by integers and division by $6$ are defined as the column multiplication and division in base 6 numerical system. The paper develops a finite automata approach for analysis of sequences $\left (\left \lfloor \xi \left (\frac{3}{2} \right)^n \right \rfloor \right)_{n \in {\mathbb Z}}$ for the words $\xi \in {\mathbb Z}^{\pm \omega}_{6}$ that have some common properties with $Z$-numbers in Mahler's $3/2$-problem. Such sequence of $Z$-words written under each other with the same digit positions in the same column is an infinite $2$-dimensional word over the alphabet ${\mathbb Z}_6$. The automata representation of the columns in the integer part of $2$-dimensional $Z$-words has the nice structural properties of the de Bruijn graphs. This way provides some sufficient conditions for the emptiness of the set of $Z$-numbers. Our approach has been initially inspirated by the proposition 2.5 in [1] where authors applies cellular automata for analysis of $\left(\left\{\xi\left(\frac{3}{2}\right)^n\right\} \right)_{n\in{\mathbb Z}}$, $\xi\in{\mathbb R}$.