Syndeticity and independent substitutions

TL;DR

This paper explores the properties of sequences generated by independent substitutions linked to abstract numeration systems, establishing conditions under which factors appear with bounded gaps and extending Cobham's theorem.

Contribution

It introduces the notion of independent substitutions based on growth order and proves a new bounded gaps property for sequences generated by such substitutions, extending Cobham's theorem.

Findings

Sequences generated by two independent substitutions with at least one exponential growth have factors with bounded gaps.

Established an analogue of Cobham's theorem for systems with polynomial and exponential growth.

Linked growth order of substitutions to combinatorial properties of generated sequences.

Abstract

We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is the following. If a sequence $x$ is generated by two independent substitutions, at least one being of exponential growth, then the factors of $x$ appearing infinitely often in $x$ appear with bounded gaps. As an application, we derive an analogue of Cobham's theorem for two independent substitutions (or abstract numeration systems) one with polynomial growth, the other being exponential.