Arithmetical subword complexity of automatic sequences

TL;DR

This paper classifies automatic sequences based on their arithmetical subword complexity, providing a complete characterization of sequences where every word appears along an arithmetic progression and deriving asymptotic formulas for their complexity.

Contribution

It offers a full classification of automatic sequences with maximal arithmetical subword complexity and general asymptotic formulas for their complexity measures.

Findings

Sequences with maximal arithmetical subword complexity are fully characterized.

An asymptotic formula for the arithmetical subword complexity of automatic sequences is established.

The results extend to polynomial subword complexity cases.

Abstract

We fully classify automatic sequences $a$ over a finite alphabet $\Omega$ with the property that each word over $\Omega$ appears is $a$ along an arithmetic progression. Using the terminology introduced by Avgustinovich, Fon-Der-Flaass and Frid, these are the automatic sequences with the maximal possible arithmetical subword complexity. More generally, we obtain an asymptotic formula for arithmetical (and even polynomial) subword complexity of a given automatic sequence $a$.