## Sums of digits, overlaps, and palindromes

Venue: | Discrete Math. & Theoret. Comput. Sci |

Citations: | 8 - 1 self |

### BibTeX

@ARTICLE{Allouche_sumsof,

author = {Jean-paul Allouche and Jeffrey Shallit},

title = {Sums of digits, overlaps, and palindromes},

journal = {Discrete Math. & Theoret. Comput. Sci},

year = {},

volume = {4},

pages = {1--10}

}

### OpenURL

### Abstract

Let ¦¨§�©��� � denote the sum of the digits in the base- � representation of �. In a celebrated paper, Thue showed that the infinite word ©� ¦ ¥ ©������������������� � is overlap-free, i.e., contains no subword of the form �������� � , where � is any finite word and � is a single symbol. Let ���� � be integers with ���� � , ���� �. In this paper, generalizing Thue’s result, we prove that the infinite word �¨§� � �� � ��©�¦¨§�©������������� � ���� � is overlap-free if and only if ���� �. We also prove that ��§¨ � � contains arbitrarily long squares (i.e., subwords of the form �� � where � is nonempty), and contains arbitrarily long palindromes if and only if ���� �.

### Citations

55 | The ubiquitious Prouhet-Thue-Morse sequence
- Allouche, Shallit
- 1999
(Show Context)
Citation Context ...ions are based on what is now called the Thue-Morse sequence ���������������s����� ¡ �����¨�����s�s�s���s�s���s�s�s�����¨��� There are many alternative ways to define this sequence (see, for example, =-=[3]-=-), one being as the fixed point, starting with � , of the � ���������smorphism � �s���s� , . One can also � define in terms of sums of digits. We ��� ����� define to be the sum of the digits in base-�... |

39 | Transcendence of Sturmian or morphic continued fractions
- Allouche, Davison, et al.
(Show Context)
Citation Context ...mes in sequences have several applications. For example, in number theory they aid in proving the transcendence of real numbers whose base expansion or continued fraction expansion have “repetitions” =-=[11, 4, 16, 2]-=-, while in statistical physics they are useful for studying the spectrum of certain discrete Schrödinger operators [9, 13, 1, 5]. 2 Some useful lemmas In this section we introduce some notation and pr... |

22 | Axel Thue’s work on repetitions in words
- Berstel
- 1992
(Show Context)
Citation Context ...me 1 Introduction At the beginning of the 20th century, the Norwegian mathematician Axel Thue initiated the study of what is now called combinatorics on words with his results on repetitions in words =-=[18, 19, 6, 8]-=-. We say a nonempty word � is a square if it can be written in the form ��� for some word � . Examples include the wordschercher in French andmurmur in English. We say that � is an overlap if it can b... |

13 | Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms
- Allouche, Zamboni
- 1998
(Show Context)
Citation Context ...mes in sequences have several applications. For example, in number theory they aid in proving the transcendence of real numbers whose base expansion or continued fraction expansion have “repetitions” =-=[11, 4, 16, 2]-=-, while in statistical physics they are useful for studying the spectrum of certain discrete Schrödinger operators [9, 13, 1, 5]. 2 Some useful lemmas In this section we introduce some notation and pr... |

8 |
Schrödinger operators with Rudin-Shapiro potentials are not palindromic
- Allouche
- 1997
(Show Context)
Citation Context ...hose base expansion or continued fraction expansion have “repetitions” [11, 4, 16, 2], while in statistical physics they are useful for studying the spectrum of certain discrete Schrödinger operators =-=[9, 13, 1, 5]-=-. 2 Some useful lemmas In this section we introduce some notation and prove some useful lemmas. Lemma 2 For any � � ��¡ £ defined by � �¢¡ £ Proof. Left to the reader. ¡s�s� , the � � sequence � � � �... |

8 | A note on palindromicity
- Baake
- 1999
(Show Context)
Citation Context ...hose base expansion or continued fraction expansion have “repetitions” [11, 4, 16, 2], while in statistical physics they are useful for studying the spectrum of certain discrete Schrödinger operators =-=[9, 13, 1, 5]-=-. 2 Some useful lemmas In this section we introduce some notation and prove some useful lemmas. Lemma 2 For any � � ��¡ £ defined by � �¢¡ £ Proof. Left to the reader. ¡s�s� , the � � sequence � � � �... |

6 | A rewriting of Fife’s theorem about overlap-free words
- Berstel
(Show Context)
Citation Context ...The Dutch chess master Max Euwe rediscovered Thue’s construction in connection with a problem about infinite chess games [10]. Fife [12] described all infinite overlap-free binary sequences; also see =-=[7]-=-. Séébold proved the beautiful and remarkable result that � is essentially the only infinite overlap-free binary sequence which is generated by iterating a morphism [17]. It is natural to wonder if Th... |