# On low-complexity bi-infinite words and their factors

• Volume: 13, Issue: 2, page 421-442
• ISSN: 1246-7405

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## Abstract

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In this paper we study bi-infinite words on two letters. We say that such a word has stiffness $k$ if the number of different subwords of length $n$ equals $n+k$ for all $n$ sufficiently large. The word is called $k$-balanced if the numbers of occurrences of the symbol a in any two subwords of the same length differ by at most $k$. In the present paper we give a complete description of the class of bi-infinite words of stiffness $k$ and show that the number of subwords of length $n$ from this class has growth order ${n}^{3}$. In the case $k=1$ we give an exact formula. We also consider the class of $k$-balanced bi-infinite words. It is well-known that the number of subwords of length $n$ from this class has growth order ${n}^{3}$ if $k=1$. In contrast, we show that the number is $\ge {2}^{n/2}$ when $k\ge 2$.

## How to cite

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Heinis, Alex. "On low-complexity bi-infinite words and their factors." Journal de théorie des nombres de Bordeaux 13.2 (2001): 421-442. <http://eudml.org/doc/248689>.

@article{Heinis2001,
abstract = {In this paper we study bi-infinite words on two letters. We say that such a word has stiffness $k$ if the number of different subwords of length $n$ equals $n + k$ for all $n$ sufficiently large. The word is called $k$-balanced if the numbers of occurrences of the symbol a in any two subwords of the same length differ by at most $k$. In the present paper we give a complete description of the class of bi-infinite words of stiffness $k$ and show that the number of subwords of length $n$ from this class has growth order $n^3$. In the case $k = 1$ we give an exact formula. We also consider the class of $k$-balanced bi-infinite words. It is well-known that the number of subwords of length $n$ from this class has growth order $n^3$ if $k = 1$. In contrast, we show that the number is $\ge 2^\{n/2\}$ when $k \ge 2$.},
author = {Heinis, Alex},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {bi-infinite words},
language = {eng},
number = {2},
pages = {421-442},
publisher = {Université Bordeaux I},
title = {On low-complexity bi-infinite words and their factors},
url = {http://eudml.org/doc/248689},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Heinis, Alex
TI - On low-complexity bi-infinite words and their factors
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 421
EP - 442
AB - In this paper we study bi-infinite words on two letters. We say that such a word has stiffness $k$ if the number of different subwords of length $n$ equals $n + k$ for all $n$ sufficiently large. The word is called $k$-balanced if the numbers of occurrences of the symbol a in any two subwords of the same length differ by at most $k$. In the present paper we give a complete description of the class of bi-infinite words of stiffness $k$ and show that the number of subwords of length $n$ from this class has growth order $n^3$. In the case $k = 1$ we give an exact formula. We also consider the class of $k$-balanced bi-infinite words. It is well-known that the number of subwords of length $n$ from this class has growth order $n^3$ if $k = 1$. In contrast, we show that the number is $\ge 2^{n/2}$ when $k \ge 2$.
LA - eng
KW - bi-infinite words
UR - http://eudml.org/doc/248689
ER -

## References

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