Subword Complexity and k-Synchronization

TL;DR

This paper investigates the properties of subword complexity functions in k-automatic sequences, demonstrating their k-synchronization and exploring related factor counting functions, with implications for understanding automatic sequences' combinatorial structure.

Contribution

It proves that subword complexity functions are k-synchronized for k-automatic sequences and extends results to primitive words and powers, contrasting with unbordered factors.

Findings

Subword complexity p_x(n) is k-synchronized for k-automatic sequences.

Results extend to counting primitive words and powers within sequences.

Unbordered factor count functions are not necessarily k-synchronized.

Abstract

We show that the subword complexity function p_x(n), which counts the number of distinct factors of length n of a sequence x, is k-synchronized in the sense of Carpi if x is k-automatic. As an application, we generalize recent results of Goldstein. We give analogous results for the number of distinct factors of length n that are primitive words or powers. In contrast, we show that the function that counts the number of unbordered factors of length n is not necessarily k-synchronized for k-automatic sequences.