Arithmetical subword complexity of automatic sequences

Abstract

We fully classify automatic sequences a over a finite alphabet Ω with the property that each word over Ω 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.