Automatic sequences based on Parry or Bertrand numeration systems

TL;DR

This paper explores the properties of automatic sequences based on Parry and Bertrand numeration systems, revealing differences from traditional k-automatic sequences in complexity and closure properties.

Contribution

It introduces the study of Parry- and Bertrand-automatic sequences, highlighting their complexity and closure property differences from classical automatic sequences.

Findings

Parry-automatic sequences have sublinear factor complexity

Bertrand-automatic sequences can have superlinear factor complexity

Closure properties differ from k-automatic sequences

Abstract

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical $k$-automatic sequences and Pisot-automatic sequences. We show that, like $k$-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for $k$-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is $U$-automatic with respect to a positional numeration system $U$ with regular language of numeration if and only if its $U$-kernel is finite.