Abstract
Two infinite words that are connected with some significant univoque numbers are studied. It is shown that their factor and palindromic complexities almost coincide with the factor and palindromic complexities of the famous Thue-Morse word.
Highlights
Two infinite words that are connected with some significant univoque numbers are studied
It is shown that their factor and palindromic complexities almost coincide with the factor and palindromic complexities of the famous Thue-Morse word
The main result of this paper is the computation of the factor and palindromic complexity of two infinite words which appear in [1] as a representation of some significant univoque numbers
Summary
The main result of this paper is the computation of the factor and palindromic complexity of two infinite words which appear in [1] as a representation of some significant univoque numbers. Komornik and Loreti showed in [2] that there is a smallest univoque number γ in the interval (1, 2) This number is transcendental [3] and is connected with the Thue-Morse word in this sense: if 1 = i>0 aiγ−i, a1a2a3 · · · = 11010011 · · · = 0−1uTM, i.e., the Thue-Morse word without the leading zero. The second one is the work of Allouche and Frougny [1] They proved that there exists a smallest univoque number in (b, b + 1) (this is proved in [5]) and they found the corresponding unique expansion of 1. These expansions and some other significant words from [1] are studied in the sequel. Here we derive both complexities directly from the definition of words which really enlighten the connection between the studied words and the Thue-Morse word
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