WORDS 2017: Combinatorics on Words pp 121-131

# On Arithmetic Index in the Generalized Thue-Morse Word

• Olga G. Parshina
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)

## 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.

## Keywords

Arithmetic index Arithmetic progression Thue-Morse word

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