aa r X i v : . [ m a t h . C O ] A p r Factor Colorings of Linearly Recurrent Words
Ville Salo
Center for Mathematical ModelingUniversity of ChileSantiago, Chile [email protected]
Ilkka T¨orm¨a
TUCS – Turku Centre for Computer ScienceUniversity of TurkuTurku, Finland [email protected]
Mathematics Subject Classifications: 68R15, 05D10
Abstract
In this short article, we study factor colorings of aperiodic linearlyrecurrent infinite words. We show that there always exists a coloringwhich does not admit a monochromatic factorization of the word intofactors of increasing lengths.
Keywords: infinite word; linearly recurrent word; factor coloring; Ramseytheory
Consider an infinite word x over a finite alphabet A , and let c be a functionfrom the finite factors of x to some finite set of colors. We can view it asa coloring of 2-subsets of the natural numbers, where the color of { i, j } for i < j is the color of the factor x [ i,j − . The infinite version of Ramsey’stheorem states that there is an infinite c -monochromatic subset of N . If we1numerate this subset as { i , i , i , . . . } , this means that x has a factorization x = x [0 ,i − x [ i ,i − x [ i ,i − · · · where every factor except the first has thesame color.Now, a natural question arises: can we get rid of the offending first factor?That is, when can we guarantee that a complete monochromatic factorizationexists? This question was raised in [6], and the first general treatment wasgiven in [2], where it was proved for several important classes of infinitewords that monochromatic factorizations need not exist. In fact, the authorsconjectured that all aperiodic words have this property.One well known class of words not considered in the study is that of fixedpoints of primitive substitutions. We consider a strictly larger class of words,the linearly recurrent ones, which have been studied in [3, 5], among others.In this short article, we show that every aperiodic linearly recurrent wordadmits a factor coloring with no monochromatic monotone factorization,that is, one where the lengths of the factors are increasing. Note that thealmost monochromatic factorization given by Ramsey’s theorem can be mademonotone, since any two adjacent factors (except the first) can be mergedtogether to produce a longer factor with the same color. In this article, the set of integers includes zero: N = { , , , . . . } . Theindexing of finite and infinite words also starts at 0. We denote by A ω theright-infinite words over an alphabet A , while A ∗ , A + , A n and A ≤ n denotewords of any finite length, length at least 1, length exactly n and length atmost n , respectively.Let x ∈ A ∗ ∪ A ω be a finite or right-infinite word, and let Fact + x, Pref + x ⊂ A + be the sets of nonempty factors and nonempty prefixes of x , respectively.A factor coloring of x is a function c : Fact + x → C , where C is a finite setof colors . A factorization of x is a (finite if x ∈ A ∗ ) sequence ( U i ) i of wordsin Fact + x such that x = Q i U i , where the product denotes concatenation,and it is monotone if | U i | ≤ | U j | for all i ≤ j . Another factorization ( V j ) j is coarser than ( U i ) i if for all j there exists i such that | V · · · V j | = | U · · · U i | .The factorization ( U i ) i is c -monochromatic if c ( U i ) = c ( U j ) holds for all i, j ,and strongly c -monochromatic if all factorizations coarser than it are also c -monochromatic. Definition 1.
For h, k ≥
1, let P ( h, k ) be the class of infinite words x ∈ A ω c : Fact + x → C with | C | ≤ k suchthat no prefix of x has a c -monochromatic factorization of length h . We alsodefine the classes P m ( h, k ), which only considers monotone factorizations,and P s ( h, k ), which only considers strongly c -monochromatic factorizations.We denote P ( k ) = P ( ∞ , k ) and P = S k ≥ P ( k ), and analogously for theother classes.Observe that we trivially have P ( h, k ) ⊂ P m ( h, k ) ⊂ P s ( h, k ) for all h, k ≥
1. Furthermore, P ( h, k ) ⊂ P ( h ′ , k ′ ) holds whenever h ≤ h ′ and k ≤ k ′ , and similarly for the other two classes. Of course, the degeneratecases h = 1 and k = 1 give rise to empty classes. Example 2.
Let x ∈ A ω be any infinite word and u ∈ A + any nonemptyfinite word. We show that u h x / ∈ P m ( h, k ) holds for all h, k ≥
1. Namely,if c : Fact + u h x → C is any factor coloring of u h x , then ( u, u, . . . , u ) (with h copies of u ) is a monotone monochromatic factorization of length h of aprefix of x .An analogous but somewhat weaker result holds for the classes P s ( h, k ).Let H ∈ N be such that every k -colored complete graph of size largerthan H contains a monochromatic induced subgraph of size h + 1; its ex-istence follows from the finite version of Ramsey’s theorem. We claim that y = u H x / ∈ P s ( h, k ), and for that, let c : Fact + y → C be a factor coloringwith | C | ≤ k . Let G be the complete graph whose vertices are the inte-gers V = { , | u | , . . . , H | u |} , and let c ′ be the edge coloring of G given by c ′ ( { i, j } ) = c ( y [ i,j − ) for all i < j ∈ V . By the definition of H , there existsa monochromatic subset W ⊂ V of size h + 1; denote W = { n , n , . . . , n h } with n i < n i +1 . Since c ′ ( { i, j } ) = c ′ ( { i − | u | , j − | u |} ) holds for all applicable i and j , we may assume n = 0. But then ( y [ n i ,n i +1 − ) ≤ i 1. But then the infinite tail x [ i , ∞ ) has the factorization ( x [ i j ,i j +1 − ) j ∈ N , which is strongly c -monochromatic, andwe choose i = i .In [2], it was proved (among other results) that all words that are notuniformly recurrent are in P , all words with full factor complexity (Fact + x = A + ) are in P (2), and all Sturmian words are in P (3). Recall that Sturmianwords are exactly the infinite words over the alphabet { , } that are balanced (the number of 1s in any two factors of the same length differs by at most1). The authors conjectured that all aperiodic infinite words are in the class P , and by the results stated above, all still unsolved cases are uniformlyrecurrent.The parameter h and the classes P m and P s were not considered in [2],and some of their basic properties are still unknown. For example, it is clearthat S h ≥ P ( h, k ) ⊂ P ( ∞ , k ) holds for all k ≥ 1, and analogously for theother classes, but we do not know whether the inclusion is strict (except inthe degenerate case k = 1).Let x ∈ A ω be an infinite word. For u ∈ Fact + x , a return to u (in x ) isa word w ∈ A + such that wu ∈ Fact + x and u occurs exactly twice in wu .The set of returns to u in x is denoted R x ( u ). We say that x is uniformlyrecurrent , if for every factor u ∈ Fact + x , the set R x ( u ) is nonempty and finite.Equivalently, every factor of x occurs infinitely many times and with boundedgaps. If x is uniformly recurrent, then for each nonempty prefix u of x , wehave a unique factorization x = r r r · · · where r i ∈ R x ( u ) for all i ∈ N .We fix some enumeration of R x ( u ), which defines a bijection Θ ux : R x ( u ) →R x ( u ), where R x ( u ) = { , , . . . , |R x ( u ) |− } . We extend Θ ux into a morphismfrom R x ( u ) ∗ to A ∗ by Θ ux ( n n · · · n k − ) = Θ ux ( n )Θ ux ( n ) · · · Θ ux ( n k − ). Thefollowing lemma can be found in [4]. Lemma 4 (Proposition 2.6 in [4]) . Let x ∈ A ω be a uniformly recurrentword, and let u, v ∈ Pref + x with | v | ≤ | u | . . The morphism Θ ux : R x ( u ) ∗ → A ∗ is injective.2. We have R x ( u ) ⊂ Θ vx ( R x ( v ) ∗ ) .3. There is a unique morphism λ uv : R x ( u ) ∗ → R x ( v ) ∗ such that Θ vx ◦ λ uv =Θ ux . A word x ∈ A ω is linearly recurrent (with constant K ) if for all u ∈ Fact + x and all w ∈ R x ( u ) we have | w | ≤ K | u | . It is periodic if x = x [ n, ∞ ) for some n ≥ 1, and aperiodic otherwise. We need the following facts about linearlyrecurrent words, which can be found in [3]. Lemma 5 (Theorem 24 in [3]) . Let x ∈ A ω be an aperiodic linearly recurrentword with constant K .1. For all u ∈ A + we have u K +1 / ∈ Fact + x .2. For all u ∈ Fact + x and w ∈ R x ( u ) we have K | w | > | u | > | w | /K 3. For all u ∈ Fact + x we have |R x ( u ) | ≤ K ( K + 1) . Important classes of linearly recurrent words include certain Sturmianwords , and all aperiodic fixed points of primitive substitutions [3]. Namely,a Sturmian word is linearly recurrent if and only if the continued fractionexpansion of its asymptotic density of 1s has bounded coefficients (see [1] fordetails). Recall that a substitution σ : A ∗ → A ∗ is primitive if there exists n ∈ N such that all a ∈ A occur in σ n ( b ) for any b ∈ A . In this section, we present our main result. Theorem 6. If x ∈ A ω is linearly recurrent with constant K and aperiodic,then x ∈ P m . More specifically, we have x ∈ P m ( K + 1 , k ) for k = 2 + P K − i =0 K i ( K + 1) i .Proof. Denote p n = x [0 ,K n − , the prefix of x of length K n , and let B = { , . . . , K ( K + 1) − } . We use B as a common index set for the returnwords of every prefix p n , since it contains R x ( p n ) for all n ∈ N . Consider aword u ∈ Fact + x . If u ∈ R x ( p n ) m for some m ∈ N , then Lemma 4 implies5hat there exists a unique word r ∈ B + such that u = Θ p n − x ( r ). We claimthat | r | < mK . This is because | w | > | p n − | /K = K n − holds for every w ∈ R x ( p n − ) by Lemma 5, so that we have Θ p n − x ( r ) > mK n − | r | , while | u | ≤ mK | p n | = mK n +1 . We denote this r by r ( u ). Finally, denote by n ( u )the unique number n ∈ N such that K n ≤ | u | < K n +1 .Using the notions defined in the above paragraph, we define the followingfactor coloring c : Fact + x → C of x , where C = { , $ } ∪ ( Z × B 1, thisimplies U K +10 ∈ Fact + x , again a contradiction.Since every fixed point of a primitive substitution is linearly recurrent,we have the following corollary. Corollary 7. If x ∈ A ω is the fixed point of an aperiodic primitive substitu-tion, then x ∈ P m . References [1] Julien Cassaigne. Recurrence in infinite words. 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