On Arithmetic Index in the Generalized Thue-Morse Word

Abstract

Let q be a positive integer. Consider an infinite word $\omega =w_0w_1w_2\cdots$ over an alphabet of cardinality q. A finite word u is called an arithmetic factor of $\omega$ if $u=w_cw_{c+d}w_{c+2d}\cdots w_{c+(|u|-1)d}$ for some choice of positive integers c and d. We call c the initial number and d the difference of u. For each such u we define its arithmetic index by $\lceil \log _q d\rceil$ where d is the least positive integer such that u occurs in $\omega$ as an arithmetic factor with difference d. In this paper we study the rate of growth of the arithmetic index of arithmetic factors of a generalization of the Thue-Morse word defined over an alphabet of prime cardinality. More precisely, we obtain upper and lower bounds for the maximum value of the arithmetic index in $\omega$ among all its arithmetic factors of length n.

This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Universite de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).