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Abelian Powers and Repetitions in Sturmian Words
[article]

2016
*
arXiv
*
pre-print

We improve on this result by studying the maximum exponents of

arXiv:1506.02797v4
fatcat:kyybzqmrxfhl3h2j6x7hnghlfu
*abelian**powers**and**abelian**repetitions*(an*abelian**repetition*is an analogue of a fractional*power*)*in**Sturmian**words*. ...*power*(resp. of an*abelian**repetition*) of*abelian*period m (the superior limits coincide for*Sturmian**words*). ... of*abelian**powers**and**repetitions**in**Sturmian**words*. ...##
###
Abelian powers and repetitions in Sturmian words

2016
*
Theoretical Computer Science
*

We improve on this result by studying the maximum exponents of

doi:10.1016/j.tcs.2016.04.039
fatcat:nxke73lqu5ctzlwt23jk7gawwe
*abelian**powers**and**abelian**repetitions*(an*abelian**repetition*is an analogue of a fractional*power*)*in**Sturmian**words*. ... denotes the maximum exponent of an*abelian**power*(resp. of an*abelian**repetition*) of*abelian*period m (the superior limits coincide for*Sturmian**words*). ... of*abelian**powers**and**repetitions**in**Sturmian**words*. ...##
###
Abelian Repetitions in Sturmian Words
[chapter]

2013
*
Lecture Notes in Computer Science
*

We investigate

doi:10.1007/978-3-642-38771-5_21
fatcat:u7br3sk5mba5bcymopm3lzezzy
*abelian**repetitions**in**Sturmian**words*. ... If km denotes the maximal exponent of an*abelian**repetition*of period m, we prove that lim sup km/m ≥ √ 5 for any*Sturmian**word*,*and*the equality holds for the Fibonacci infinite*word*. ... An*abelian**power*is an*abelian**repetition**in*which u 0 = u j = ε. We call u 0*and*u j the head*and*the tail of the*abelian**repetition*, respectively. ...##
###
Abelian Repetitions in Sturmian Words
[article]

2013
*
arXiv
*
pre-print

We investigate

arXiv:1209.6013v3
fatcat:3n4lhospzfhptjz7lxc56sdngq
*abelian**repetitions**in**Sturmian**words*. ... We prove that*in*any*Sturmian**word*the superior limit of the ratio between the maximal exponent of an*abelian**repetition*of period m*and*m is a number ≥√(5),*and*the equality holds for the Fibonacci infinite ... Similar techniques were used*in*[40] to derive nice results on*abelian**powers*on*Sturmian**words*, that are*repetitions*with empty head*and*tail. ...##
###
Page 2779 of Mathematical Reviews Vol. , Issue 99d
[page]

1991
*
Mathematical Reviews
*

We study the minimal proportion of one letter

*in*an xth-*power*- free binary*word*as a function of x*and*prove,*in*particular, that this function is discontinuous.” ... The AR problem— to find all*abelian**repetitions*—generalizes the R problem; it is conjectured that the AR problem requires @(n7) time*in*the worst case. ...##
###
Words with the Maximum Number of Abelian Squares
[article]

2015
*
arXiv
*
pre-print

An

arXiv:1506.03562v1
fatcat:i2fgs6ze6zh6jfpuau3e5dpxqq
*abelian*square is the concatenation of two*words*that are anagrams of one another. A*word*of length n can contain Θ(n^2) distinct factors that are*abelian*squares. ... We study infinite*words*such that the number of*abelian*square factors of length n grows quadratically with n. ...*In*this case, we prove that if a*Sturmian**word*is β-*power*free for some β ≥ 2 (that is, does not contain*repetitions*of order β or higher), then it is uniformly*abelian*-square rich. ...##
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Abelian Properties of Words (Extended abstract)
[article]

2009
*
arXiv
*
pre-print

We also investigate

arXiv:0904.2925v1
fatcat:tto6ecib2berlgwgtgwm4rmx2e
*abelian**repetitions**in**words**and*show that any infinite*word*with bounded*abelian*complexity contains*abelian*k-*powers*for every positive integer k. ...*In*this paper we investigate various*abelian*properties of*words*including*abelian*complexity,*and**abelian**powers*. ... Section 9 contains a detailed study of*abelian**powers**in**Sturmian**words*, the Thue-Morse*word**and*the Tribonacci*word*. ...##
###
Abelian-square-rich words

2017
*
Theoretical Computer Science
*

As for

doi:10.1016/j.tcs.2017.02.012
fatcat:kvuemu6aw5djvkvyzv7hqghcce
*Sturmian**words*, we prove that a*Sturmian**word*s_α of angle α is uniformly*abelian*-square-rich if*and*only if the irrational α has bounded partial quotients, that is, if*and*only if s_α has bounded ...*in*n;*and*uniformly*abelian*-square-rich if every factor of w contains a number of distinct*abelian*-square factors that is proportional to the square of its length. ... A*word*w is a k-*power*(also called a*repetition*of order k), for an integer k ≥ 2, if there exists a nonempty*word*u such that w = u k . A 2-*power*is called a square. ...##
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On a generalization of Abelian equivalence and complexity of infinite words
[article]

2013
*
arXiv
*
pre-print

Moreover if ω is aperiodic, then P^(k)_ω(n)=q^k(n) if

arXiv:1301.5104v1
fatcat:mro2owfferhnbc5ok6niu5iwya
*and*only if ω is*Sturmian*. We also study k-*Abelian*complexity*in*connection with*repetitions**in**words*. ... Two finite*words*u*and*v*in*A^* are said to be k-*Abelian*equivalent if for all x∈ A^* of length less than or equal to k, the number of occurrences of x*in*u is equal to the number of occurrences of x*in*... Recently k-*Abelian*equivalence has been studied*in*the context of avoidance of*repetitions**in**words*(see the discussion at the beginning of §5 on k-*Abelian**powers*). ...##
###
Abelian Combinatorics on Words: a Survey
[article]

2022
*
arXiv
*
pre-print

We survey known results

arXiv:2207.09937v1
fatcat:equyjxuidvewzpsna5yrluhpom
*and*open problems*in**abelian*combinatorics on*words*. ...*In*the past few years, there was a lot of research on*abelian*analogues of classical definitions*and*properties*in*combinatorics on*words*. This survey aims to gather these results. ...*Abelian**powers**in**Sturmian**words*Richomme, Saari*and*Zamboni [114] proved that*in*every*Sturmian**word*, for any position*and*for every positive integer k, there is an*abelian*k-*power*starting at that ...##
###
Abelian complexity of minimal subshifts

2010
*
Journal of the London Mathematical Society
*

*In*the case of

*Sturmian*

*words*we prove something stronger: For every

*Sturmian*

*word*w

*and*positive integer k, each sufficiently long factor of w begins

*in*an

*Abelian*k-

*power*. ... Rauzy by exhibiting a class of

*words*whose

*Abelian*complexity is everywhere equal to 3. We also investigate links between

*Abelian*complexity

*and*the existence of

*Abelian*

*powers*. ...

*Abelian*

*repetitions*

*in*

*Sturmian*

*words*

*In*this section, we prove the following theorem, thereby answering Open Problem 2

*in*the affirmative

*in*the case of

*Sturmian*

*words*. Theorem 6.1. ...

##
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Everywhere α-repetitive sequences and Sturmian words

2010
*
European journal of combinatorics (Print)
*

*In*this paper, we study optimal 2-

*repetitive*sequences

*and*optimal 2 + -

*repetitive*sequences,

*and*show that

*Sturmian*

*words*belong to both classes. ... Local constraints on an infinite sequence that imply global regularity are of general interest

*in*combinatorics on

*words*. We consider this topic by studying everywhere α-

*repetitive*sequences. ... Karhumäki for several discussions

*and*for his encouragement to pursue investigating the topic of this paper. Thanks also to J. Cassaigne

*and*D. ...

##
###
Abelian periods of factors of Sturmian words
[article]

2020
*
arXiv
*
pre-print

A characterization of the Fibonacci

arXiv:1905.06138v3
fatcat:gzursxyhxzfepbjiwtgignoc7y
*word**in*terms of its*abelian*period set is obtained as a corollary. ... We study the*abelian*period sets of*Sturmian**words*, which are codings of irrational rotations on a one-dimensional torus. ...*Abelian**Powers**in**Sturmian**Words*The starting point of the study of*abelian*equivalence*in**Sturmian**words*is the following result stating that factors of length n of a*Sturmian**word*belong to exactly two ...##
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All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words

2020
*
International Symposium on Mathematical Foundations of Computer Science
*

We consider

doi:10.4230/lipics.mfcs.2020.79
dblp:conf/mfcs/PeltomakiW20
fatcat:izqxtb4kr5cnfg4hjmneuwex5u
*repetitions**in*infinite*words*by making a novel inquiry to the maximum eventual growth rate of the exponents of*abelian**powers*occurring*in*an infinite*word*. ... As a consequence, we obtain that every nonnegative real number is the critical*abelian*exponent of some infinite binary*word*. ... The notion is particularly suitable for exponents of*abelian**powers**in**Sturmian**words**and*leads to surprising results. ...##
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On the Contribution of WORDS to the Field of Combinatorics on Words
[chapter]

2015
*
Lecture Notes in Computer Science
*

This prologue summarizes the history of the conference

doi:10.1007/978-3-319-23660-5_3
fatcat:i6xrc7lnobdkrhwq3xib3rjvga
*WORDS*,*and*the contributions which were presented, our goal being to testify how the conference may be embedded*in*the development of the field of ... Combinatorics on*Words*. ... 2001), Fabio Burderi (*WORDS*2011)*and*Dominique Perrin (*WORDS*2015). Extensions of the classical concept of*words*were the subject of a lot of presentations. ...
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