De Bruijn graphs and powers of $3/2$

zbMATH Open1475.68161arXiv1811.02254MaRDI QIDQ5158424

Author name not available (Why is that?)

Publication date: 22 October 2021

Abstract: In this paper we consider the set

{m a t h b b Z}_{6}^{p m o m e g a}

of two-way infinite words

x i

over the alphabet

0, 1, 2, 3, 4, 5

with the integer left part

l f l o o r x i f l o o r

and the fractional right part

x i

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

l e f t (l e f t l f l o o r x i l e f t (f r a c 32 i g h t)^{n} i g h t f l o o r i g h t)_{n i n m a t h b b Z}

for the words

x i i n {m a t h b b Z}_{6}^{p m o m e g a}

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

{m a t h b b 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

l e f t (l e f t x i l e f t (f r a c 32 i g h t)^{n} i g h t i g h t)_{n i n m a t h b b Z}

,

x i i n m a t h b b R

.

Full work available at URL: https://arxiv.org/abs/1811.02254

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