On possible growths of arithmetical complexity

Frid, Anna E.

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 3, pp. 443-458.

The arithmetical complexity of infinite words, defined by Avgustinovich, Fon-Der-Flaass and the author in 2000, is the number of words of length $n$ which occur in the arithmetical subsequences of the infinite word. This is one of the modifications of the classical function of subword complexity, which is equal to the number of factors of the infinite word of length $n$. In this paper, we show that the orders of growth of the arithmetical complexity can behave as many sub-polynomial functions. More precisely, for each sequence $u$ of subword complexity $f_u\left(n\right)$ and for each prime $p\ge 3$ we build a Toeplitz word on the same alphabet whose arithmetical complexity is $a\left(n\right)=\Theta \left(n{f}_u\left(\lceil {log}_{p}n\rceil \right)\right)$.

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