Distribution modulo 1 and the lexicographic world
aa r X i v : . [ m a t h . N T ] J a n DISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD
JEAN-PAUL ALLOUCHE AND AMY GLEN
In honour of Paulo Ribenboim on the occasion of his 80th birthday A BSTRACT . We give a complete description of the minimal intervals containing all fractional parts { ξ n } , for some positive real number ξ , and for all n ≥ .
1. I
NTRODUCTION
In the paper [20] Mahler defined the set of Z -numbers by (cid:26) ξ ∈ R , ξ > , ∀ n ≥ , ≤ (cid:26) ξ (cid:18) (cid:19) n (cid:27) < (cid:27) where { z } is the fractional part of the real number z . Mahler proved that this set is at most countable.It is still an open problem to prove that this set is actually empty. More generally, given a real number α > and an interval ( x, y ) ⊂ (0 , one can ask whether there exists ξ > such that, for all n ≥ , we have x ≤ { ξα n } < y (or the variant x ≤ { ξα n } ≤ y ). Flatto, Lagarias, and Pollington[14, Theorem 1.4] proved that, if α = p/q with p, q coprime integers and p > q ≥ , then anyinterval ( x, y ) such that for some ξ > , one has that { ξ ( p/q ) n } ∈ ( x, y ) for all n ≥ , must satisfy y − x ≥ /p . Recently Bugeaud and Dubickas [8] characterized irrational numbers ξ such that fora fixed integer b ≥ all the fractional parts { ξb n } belong to a closed interval of length /b . Beforestating their theorem we need a definition. Definition 1.
Given two real numbers α and ρ with α ≥ , we denote by s α,ρ := ( s α,ρ ( n )) n ≥ and s ′ α,ρ := ( s ′ α,ρ ( n )) n ≥ the sequences defined by s α,ρ ( n ) = ⌊ ( n + 1) α + ρ ⌋ − ⌊ nα + ρ ⌋ and s ′ α,ρ ( n ) = ⌈ ( n + 1) α + ρ ⌉ − ⌈ nα + ρ ⌉ for n ≥ , where ⌊ x ⌋ denotes the greatest integer ≤ x and ⌈ x ⌉ denotes the least integer ≥ x . The sequences s α,ρ and s ′ α,ρ are called Sturmian sequences if α is irrational, and periodic balanced sequences if α is rational. Furthermore, if α = ρ , these sequences are called characteristic Sturmian sequences or characteristic periodic balanced sequences, according to whether or not α is irrational.We observe that, if α is not an integer, then for all n ≥ , ⌊ α ⌋ ≤ s α,ρ ( n ) ≤ ⌊ α ⌋ + 1 and ⌈ α ⌉ − ≤ s ′ α,ρ ( n ) ≤ ⌈ α ⌉ , where ⌈ α ⌉ − ⌊ α ⌋ and ⌈ α ⌉ = ⌊ α ⌋ + 1 . On the other hand, if α is an integer, then s α,ρ = s ′ α,ρ and α ≤ s α,ρ ( n ) ≤ α + 1 for all n ≥ . Accordingly, the sequences s α,ρ and s ′ α,ρ take their values in Mathematics Subject Classification.
Key words and phrases. distribution modulo ; Z -numbers; lexicographic world; Sturmian sequences; balanced se-quences; central words; palindromic closure. the “alphabet” { k, k + 1 } where k = ⌊ α ⌋ . The classical definition of Sturmian sequences with valuesin { , } is thus obtained by subtracting ⌊ α ⌋ from each of the terms in the sequences s α,ρ and s ′ α,ρ .Alternatively, one may restrict α to the interval (0 , . Hereafter, if the alphabet is not mentioned, it isunderstood that the sequences are over { , } . We may also assume that ρ ∈ [0 , or ρ ∈ (0 , since s α,ρ = s α,ρ ′ and s ′ α,ρ = s ′ α,ρ ′ for any two real numbers ρ , ρ ′ such that ρ − ρ ′ is an integer. Example 2.
Taking α = ρ = (3 − √ / , we get the well-known (binary) Fibonacci sequence · · · . Remark . Note that if α is irrational, then the (Sturmian) sequences s α,ρ and s ′ α,ρ are aperiodic (i.e.,not eventually periodic), whereas if α is rational, the sequences s α,ρ and s ′ α,ρ are (purely) periodic.(See for instance [19, Lemma 2.14].) This justifies the use of “periodic” in the name of such sequencesin the rational case. The reason for being called “balanced” is explained in Section 3.1.Let T denote the shift map on sequences, defined as follows: if s := ( s n ) n ≥ , then T ( s ) = T (( s n ) n ≥ ) := ( s n +1 ) n ≥ . The main result in [8] reads as follows. Theorem 4 (Bugeaud-Dubickas) . Let b ≥ be an integer and let ξ be an irrational number. Thenthe numbers { ξb n } cannot all lie in an interval of length < /b . Furthermore there exists a closedinterval I of length /b containing the numbers { ξb n } for all n ≥ if and only if the sequence ofbase b digits of the fractional part of ξ is a Sturmian sequence s on the alphabet { k, k + 1 } for some k ∈ { , , . . . , b − } . If this is the case, then ξ is transcendental, and the interval I is semi-open. Itis open unless there exists an integer j ≥ such that T j ( s ) is a characteristic Sturmian sequence onthe alphabet { k, k + 1 } . The purpose of this paper is to give a complete description of the minimal intervals containing allfractional parts { ξ n } for some positive real number ξ , and for all n ≥ . More precisely, inspiredby the definition of the lexicographic world (see Section 2.2), let us define a function F on [0 , asfollows. Definition 5.
For all x ∈ [0 , , let S x := { ξ ∈ R , ξ > , ∀ n ≥ , x ≤ { ξ n } < } and let F : [0 , → [0 , be the function defined by: F ( x ) = ( inf { y ∈ [0 , , ∃ ξ > , ∀ n ≥ , x ≤ { ξ n } ≤ y } if S x = ∅ , if S x = ∅ . Remark . From Bugeaud-Dubickas’ result for b = 2 , we deduce the following two facts. • For x ∈ [ , , there does not exist an irrational number ξ > such that x ≤ { ξ n } < forall n ≥ . Nor does there exist a rational number ξ > such that x ≤ { ξ n } < for all n ≥ . (This can be seen by considering, for instance, the base expansion of the fractionalpart of ξ for any rational number ξ ≥ x .) Hence, F ( x ) = 1 for all x ∈ [ , . • If ξ > is an irrational real number, then there exists a real number x ∈ [0 , ) such that all thefractional parts { ξ n } belong to the interval [ x, x + ] if and only if the base expansion of thefractional part of ξ is a Sturmian sequence. Furthermore, for any such x , one has F ( x ) = x + .Note that it follows from our main theorem (see Theorem 7 later) that ≤ F ( x ) < for x ∈ [0 , ) . ISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD 3
Before stating our main theorem, let us note that the sequences s α,ρ and s ′ α,ρ (given in Definition 1)are said to have slope α and intercept ρ , in view of their geometric realization as approximations tothe line y = αx + ρ (called lower and upper mechanical words in [19, Chapter 2]). From now on,we will assume that α and ρ are in the interval [0 , , in which case the sequences s α,ρ and s ′ α,ρ taketheir values in { , } . If α is irrational, then we have s α,α = s ′ α,α , denoted by c α . We also have s , = s ′ , = 0 ∞ and s , = s ′ , = 1 ∞ , denoted by c and c , respectively. In these cases, thesequence c α is the unique characteristic Sturmian sequence of slope α in { , } N . On the other hand,if α ∈ (0 , is rational, then the characteristic periodic balanced sequences of slope α , namely s α,α and s ′ α,α , are distinct sequences in { , } N containing both ’s and ’s. More precisely, let us supposethat α = p/q ∈ (0 , with gcd( p, q ) = 1 . Then by considering the prefix of length q of each of thesequences s p/q, and s ′ p/q, , we find that there exists a unique word w p,q of length q − such that s p/q, = (0 w p,q ∞ and s ′ p/q, = (1 w p,q ∞ where v ∞ denotes the periodic sequence vvvv · · · for a given word v . (See for instance [19, pg. 59].)Hence, the two characteristic periodic balanced sequences of slope p/q in { , } N are given by s p/q,p/q = T ( s p/q, ) = ( w p,q ∞ and s ′ p/q,p/q = T ( s ′ p/q, ) = ( w p,q ∞ . The words w p,q are often referred to as central words in the literature; they hold a special place inthe rich theory of Sturmian sequences (see, e.g., [19, Chapter 2]). For instance, it follows from thework of de Luca and Mignosi [11, 12] that central words coincide with the palindromic prefixes ofcharacteristic Sturmian sequences (see Section 3.3).Given a sequence s ∈ { , } N , let r ( s ) denote the real number whose sequence of base digits isgiven by s . Our main number-theoretical result reads as follows. Theorem 7.
Let x be a real number in [0 , . (i) If x ≥ , then F ( x ) = 1 . (ii) If x = 0 , then F ( x ) = 0 . (iii) If x ∈ (0 , ) and if the base expansion of x is given by a characteristic Sturmian sequence,then F ( x ) = x + . Furthermore, F ( x ) is the unique real number in [0 , that has a Sturmianbase expansion and satisfies x ≤ { F ( x )2 k } ≤ F ( x ) for all k ≥ . (iv) If x ∈ (0 , ) and if the base expansion of x is given by a characteristic periodic balancedsequence of slope p/q ∈ (0 , with gcd( p, q ) = 1 , then F ( x ) is the rational number whosebase expansion is given by the periodic balanced sequence s ′ p/q, = (1 w p,q ∞ , in whichcase F ( x ) ≤ x + . (v) In all other cases, F ( x ) can be explicitly computed: it is equal to the rational number whosebase expansion is given by a (unique) periodic balanced sequence s ′ p/q, = (1 w p,q ∞ where p , q are coprime integers with < p < q such that r (( w p,q ∞ ) < x Remark . It is known (see [13]) that real numbers having a Sturmian base expansion are transcen-dental. As a consequence of Theorem 7, we deduce that, if x is an algebraic real number in [0 , ) ,then F ( x ) is rational. 2. T HE COMBINATORIAL APPROACH The main tool used by Bugeaud and Dubickas is combinatorics on words : real numbers are replacedby their base b expansion, and inequalities between real numbers are transformed into (lexicographic)inequalities between infinite sequences representing their base b expansions. We will establish a the-orem of combinatorial flavour (Theorem 13), whose translation into a number-theoretical statementis exactly Theorem 7 above. A method for computing F ( x ) in Case (v) of Theorem 7 is given inSection 6.2.1. Two combinatorial theorems. It happens that the case b = 2 of Bugeaud-Dubickas’ theoremwas already proved by Veerman in [25, 26]. The combinatorial result proved by Veerman, and byBugeaud-Dubickas is stated (and strengthened) in Theorems 9 and 10 below. Theorem 9. An aperiodic sequence s := ( s n ) n ≥ on { , } is Sturmian if and only if there exists asequence u := ( u n ) n ≥ on { , } such that u ≤ T k ( s ) ≤ u for all k ≥ . Moreover, u is theunique characteristic Sturmian sequence with the same slope as s , and we have u = inf { T k ( s ) , k ≥ } and u = sup { T k ( s ) , k ≥ } . Theorem 10. An aperiodic sequence u on { , } is a characteristic Sturmian sequence if and only if,for all k ≥ , u < T k ( u ) < u . Furthermore, we have u = inf { T k ( u ) , k ≥ } and u = sup { T k ( u ) , k ≥ } . The lexicographic world. As discussed in [1], the results in Theorems 9 and 10 have beenrediscovered several times since the work of Veerman in the mid-late 80’s. One of the presentationsof these statements is due to Gan [16]. It is based on the lexicographic(al) world , which seems to havebeen introduced in 2000, in a preprint version of [18].For any two sequences x , y ∈ { , } N , define the set Σ x , y := { s ∈ { , } N , ∀ k ≥ , x ≤ T k ( s ) ≤ y } , where ≤ denotes the lexicographic order on { , } N induced by < . The lexicographic world L isdefined by L := { ( x , y ) ∈ { , } N × { , } N , Σ x , y = ∅} . Moreover, by [16, Lemma 2.1], we have L = { ( u , v ) ∈ { , } N × { , } N , v ≥ φ ( u ) } , where φ : { , } N → { , } N is the map defined by φ ( x ) := inf { y ∈ { , } N , Σ x , y = ∅} . Trivially, φ (1 x ) = 1 ∞ = 111 · · · for any sequence x ∈ { , } N . ISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD 5 In [16], Gan showed that for any sequence u ∈ { , } N , the set Σ u , u is not empty, i.e., thereexists a sequence s ∈ { , } N such that u ≤ T k ( s ) ≤ u for all k ≥ (see [16, Lemma 4.2]). Fur-thermore, the sequence φ (0 u ) has the foregoing property (by [16, Theorem 3.4]) and it is a Sturmianor periodic balanced sequence with the property that T k ( φ (0 u )) ≤ φ (0 u ) for all k ≥ (see [16,Theorem 4.6]). Moreover, by [16, Lemma 5.4], the set Σ u , u contains a unique Sturmian or periodicbalanced sequence satisfying T k ( s ) ≤ s for all k ≥ . We deduce from these remarks that, for anysequence s ∈ { , } N , if s = φ (0 u ) for some sequence u ∈ { , } N , then s is the unique Sturmian orperiodic balanced sequence satisfying u ≤ T k ( s ) ≤ u and T k ( s ) ≤ s for all k ≥ . The converseof this statement also holds by [16, Corollary 5.6]. These observations establish Gan’s main theorem(see below), which shows in particular that any element in the image of φ is a Sturmian or periodicbalanced sequence in { , } N (and such sequences are the lexicographically greatest amongst theirshifts). Theorem 11. [16, Theorem 1.1] For any sequence s ∈ { , } N , the following conditions are equiva-lent. (i) s = φ (0 u ) for some sequence u ∈ { , } N . (ii) s is the unique Sturmian or periodic balanced sequence satisfying u ≤ T k ( s ) ≤ u and T k ( s ) ≤ s for all k ≥ .Note. “Sturmian” in Gan’s paper corresponds to what is called here (and classically) “Sturmian orperiodic balanced”. Remark . It is well known that the closure of the shift-orbit of a characteristic Sturmian sequence s (i.e., the closure of { T k ( s ) , k ≥ } , denoted by O ( s ) ) is precisely the set of all Sturmian sequenceshaving the same slope as s (see for instance [19, Propositions 2.1.25 and 2.1.18], or [21]). In view ofthis fact, Gan’s result can be strengthened using Theorem 9, as follows. If the sequence s := φ (0 u ) is Sturmian, then u is the unique characteristic Sturmian sequence in O ( s ) , in which case s = 1 u .We will further strengthen Gan’s result by describing φ (0 u ) for any given sequence u ∈ { , } N .In particular, we will show that when u contains both ’s and ’s and is not a characteristic Sturmiansequence, there exists a unique pair of characteristic periodic balanced sequences s and s ′ of (rational)slope p/q ∈ (0 , with gcd( p, q ) = 1 , such that s ′ ≤ u ≤ s , in which case φ (0 u ) = 1 s ′ . Moreover,the sequences s , s ′ can be explicitly determined in terms of u .With the same notation as in the Introduction, our main combinatorial theorem reads as follows. Theorem 13. Let u be a sequence in { , } N . (i) φ (1 u ) = 1 ∞ . (ii) If u ∈ { ∞ , ∞ } , then φ (0 u ) = u . (iii) If u is a characteristic Sturmian sequence, then φ (0 u ) = 1 u . Furthermore, u is the uniqueSturmian sequence in { , } N satisfying u ≤ T k (1 u ) ≤ u for all k ≥ . (iv) If u is a characteristic periodic balanced sequence of rational slope p/q ∈ (0 , with gcd( p, q ) = 1 , then φ (0 u ) = s ′ p/q, = (1 w p,q ∞ . (v) If u does not take any of the forms given in parts (ii)–(iv) , then there exists a unique pair ofcoprime integers p , q with < p < q such that ( w p,q ∞ < u < ( w p,q ∞ , in which case φ (0 u ) = s ′ p/q, = (1 w p,q ∞ . JEAN-PAUL ALLOUCHE AND AMY GLEN Moreover, in cases (iv) and (v), φ (0 u ) is the unique periodic balanced sequence in { , } N satisfying u ≤ T k ( φ (0 u )) ≤ φ (0 u ) for all k ≥ . In the next section, we will recall some generalities about Sturmian and periodic balanced se-quences. (For more on Sturmian sequences, the reader can consult, e.g., [19, Chapter 2].) The proof ofTheorem 13 is given in Section 4, and a corollary is stated in Section 5. Lastly, in Section 6, we showhow to determine the “central word” w p,q such that φ (0 u ) = (1 w p,q ∞ for any “generic” sequence u falling into Case (v) of Theorem 13 above.3. S TURMIAN & PERIODIC BALANCED SEQUENCES In what follows, we will use the following notation and terminology from combinatorics on words(see, e.g., [19]). Let w = x x · · · x m be a word over a finite non-empty alphabet A (where each x i is a letter in A ). The length of w , denoted by | w | , is equal to m . The empty word is the uniqueword of length , denoted by ε . The number of occurrences of a letter x in w is denoted by | w | x . The reversal of w is defined by ˜ w = x m · · · x x , and by convention ε = ˜ ε . If w = ˜ w , then w is calleda palindrome . An integer ℓ ≥ is said to be a period of w if, for all i , j with ≤ i, j ≤ m , i ≡ j (mod ℓ ) implies x i = x j . Note that any integer ℓ ≥ | w | is a period of w with this definition. The word w is said to be primitive if it is not a power of a shorter word, i.e., if w = u n implies n = 1 . A finiteword z is said to be a factor of w if z = x i x i +1 · · · x j for some i, j with ≤ i ≤ j ≤ m . Similarly, a factor of a sequence s := s s s s · · · is any finite word of the form s i s i +1 · · · s j with i ≤ j .Recall from the Introduction that the shift map T is defined on sequences as follows: if s :=( s n ) n ≥ then T ( s ) = T (( s n ) n ≥ ) := ( s n +1 ) n ≥ . This operator naturally extends to finite words as a circular shift by defining T ( xw ) = wx for any letter x and finite word w .Under the operation of concatenation, the set A ∗ of all finite words over A is a free monoid withidentity element ε and set of generators A . If x is a letter, then we use x ∗ to denote { x } ∗ , the set of allfinite powers of x . From now on, all words and sequences will be over the alphabet { , } .3.1. Balanced sequences. All Sturmian sequences are “balanced” in the following sense (see forinstance [22, 10, 5, 6, 19]). Definition 14. A finite word or sequence w over { , } is said to be balanced if, for any two factors u , v of w with | u | = | v | , we have || u | − | v | | ≤ or equivalently || u | − | v | | ≤ .Recall from Remark 3 that Sturmian sequences are aperiodic. Morse, Hedlund, and Coven [22, 10]proved that the Sturmian sequences are precisely the aperiodic balanced sequences on two letters(also see [19, Theorem 2.1.3]). “Periodic balanced sequences” (as specified in Definition 1) are alsobalanced in the sense of the above definition (which justifies their name); moreover, they constitutethe set of all periodic balanced sequences on two letters (see [19, Lemma 2.1.15] or [24]).3.2. Characteristic Sturmian sequences. In [11], characteristic Sturmian sequences were character-ized using iterated palindromic closure, defined as follows. The palindromic (right-)closure of a finiteword w , denote by w (+) , is the (unique) shortest palindrome beginning with w . That is, if w = uv where v is the longest palindromic suffix of w , then w (+) := uv ˜ u . For example, (011) (+) = 0110 .The iterated palindromic closure function , denote by P al , is defined by iteration of the palindromicright-closure operator (see, e.g., [17]). More precisely, P al is defined recursively as follows. Set ISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD 7 P al ( ε ) = ε , and for any word w and letter x , define P al ( wx ) := ( P al ( w ) x ) (+) . For example, P al (011) = ( P al (01)1) (+) = (0101) (+) = 01010 . Note that P al is injective; and moreover, it isclear from the definition that P al ( w ) is a prefix of P al ( wx ) for any word w and letter x . Hence, if v is a prefix of w , then P al ( v ) is a prefix of P al ( w ) .The following theorem provides a combinatorial description of characteristic Sturmian sequencesin terms of P al . Theorem 15. [11] For any sequence s ∈ { , } N , the following properties are equivalent. (i) s is a characteristic Sturmian sequence. (ii) There exists a (unique) sequence ∆ := x x x x . . . ∈ { , } N \ ( { , } ∗ ∞ ∪ { , } ∗ ∞ ) (i.e., not eventually constant), called the directive sequence of s , such that s = lim n →∞ P al ( x x x · · · x n ) = P al (∆) . Example 16. Recall from Example 2 that the (binary) Fibonacci sequence f = 01001010010 · · · isthe characteristic Sturmian sequence c α with α = (3 − √ / ; it has directive sequence (01) ∞ . Thatis: f = P al (0101 · · · ) = 01001010010 · · · , where the underlined letters indicate at which points palindromic closure is applied. Note the simplecontinued fraction expansion of α = (3 − √ / is [0; 2 , , , , . . . ] . More generally, if α ∈ (0 , isan irrational number with simple continued fraction expansion [0; d + 1 , d , d , d , . . . ] where d ≥ and all other d i ≥ , then c α = P al (0 d d d d · · · ) (see [15, 7] and also [2, pg. 206]).3.3. Characteristic periodic balanced sequences. We will now recall some known combinatorialdescriptions of the characteristic periodic balanced sequences in { , } N (see Proposition 17 and Re-mark 18 below).Let us first recall from the Introduction that the characteristic balanced sequences of slopes and are c = 0 ∞ and c = 1 ∞ , respectively. For all other rational slopes p/q ∈ (0 , with gcd( p, q ) = 1 ,there are exactly two characteristic periodic balanced sequences of slope p/q , given by s p/q,p/q = T ( s p/q, ) = ( w p,q ∞ and s ′ p/q,p/q = T ( s ′ p/q, ) = ( w p,q ∞ , where w p,q is a word of length q − in { , } ∗ . For example, with p = 2 and q = 5 , we obtain thefollowing two characteristic periodic balanced sequences of slope / : s / , / = (01010) ∞ and s ′ / , / = (01001) ∞ where w , = 010 .Notice that w , is a palindrome and | w , | = | w , | = 2 = p . More generally, one can verifythat all words w p,q are palindromes and | w p,q | = | w p,q | = p . Furthermore, the words w p,q and w p,q (which have length q ) are primitive since gcd( p, q ) = 1 . Hereafter, the word w p,q will becalled the central word of slope p/q ; it is the unique central word of length q − containing p − occurrences of . Note. The set of all central words of slope p/q ∈ (0 , (where p , q are coprime integers) coincideswith the family of “central words” in { , } ∗ as defined in [19, Chapter 2] (in particular, see [19,Theorem 2.2.11 and Proposition 2.2.12]). JEAN-PAUL ALLOUCHE AND AMY GLEN The following proposition collects together some equivalent definitions of central words. For manymore, see the nice survey [4]. Proposition 17. For any word w ∈ { , } ∗ , the following properties are equivalent. (i) w is a central word. (ii) w and w are balanced [12] . (iii) w = P al ( v ) for some word v ∈ { , } ∗ [12, 11] . (iv) w has two periods ℓ , m such that gcd( ℓ, m ) = 1 and | w | = ℓ + m − [10, 12] . (v) w ∈ ∗ ∪ ∗ ∪ ( P ∩ P P ) where P is the set of all palindromes in { , } ∗ [12] . (vi) w ∈ ∗ ∪ ∗ , or there exists a unique pair of words w , w ∈ { , } ∗ such that w satisfies theequation w = w w = w w [12, 11] .Moreover, in part (vi) , w and w are central words, ℓ := | w | + 2 and ℓ := | w | + 2 are coprimeperiods of w , and min { ℓ , ℓ } is the minimal period of w [9] .Note. P ∩ ( P P ) = P ∩ ( P P ) .Furthermore, by [19, Proposition 2.2.12], the central word w p,q of slope p/q is the central wordwith coprime periods ℓ , m where ℓ + m = q and mp ≡ q ) . For example, the central word w , = 010 has coprime periods ℓ = 2 and m = 3 where q and mp = 6 ≡ . Alsonote that w ,q = 0 q − and w q − ,q = 1 q − ; in particular w , = ε . Remark . Let p , q be coprime integers with < p < q . Then p/q has two distinct simple continuedfraction expansions: p/q = [0; d + 1 , . . . , d n , 1] = [0; d + 1 , . . . , d n + 1] where d ≥ and all other d i ≥ . It is known (see, e.g., [4, Proposition 27]) that the word v ∈ { , } ∗ such that w p,q = P al ( v ) takes the form v = 0 d d d · · · x d n where x = 0 if n is odd and x = 1 if n is even. For example, / , , 1] = [0; 2 , and w , = 010 = P al (01) . Moreover, as in thecase of characteristic Sturmian sequences (see Theorem 15 and Example 16), the two characteristicperiodic balanced sequences of slope p/q can be obtained by iterated palindromic closure. Moreprecisely, with the above notation, we have ( w p,q xy ) ∞ = P al (0 d d · · · x d n +1 y ∞ ) and ( w p,q yx ) ∞ = P al (0 d d · · · x d n yx ∞ ) where { x, y } = { , } . For example, ( w , ∞ = (01010) ∞ = P al (0110 ∞ ) and ( w , ∞ = (01001) ∞ = P al (0101 ∞ ) . In [23] Pirillo proved that a word w ∈ { , } ∗ is a palindromic prefix of some characteristic Stur-mian sequence in { , } N , i.e., w = P al ( v ) for some v ∈ { , } ∗ (see Theorem 15) if and only if w is a circular shift of w . From this fact and Proposition 17, we thus deduce the following result. Proposition 19. A word w ∈ { , } ∗ is central if and only if w is a circular shift of w . Consequently, for any two coprime integers p , q with < p < q , the two characteristic periodicbalanced sequences of slope p/q , namely s p/q,p/q = ( w p,q ∞ and s ′ p/q,p/q = ( w p,q ∞ , are shiftsof each other, and therefore they have the same set of factors. ISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD 9 Remark . Recall from Remark 12 that the closure of the shift-orbit of a characteristic Sturmiansequence s is precisely the set of all Sturmian sequences having the same slope as s . Moreover, the(Sturmian) sequences in O ( s ) are exactly the sequences that have the same set of factors as s (seefor instance [19, Propositions 2.1.25 and 2.1.18], or [21]). Likewise, the shift-orbit of a characteristicperiodic balanced sequence u consists of all the periodic balanced sequences with the same set offactors (and also the same slope) as u . However, in contrast to the aperiodic case, we deduce fromProposition 19 that, if u is a characteristic periodic balanced sequence containing both ’s and ’s,then O ( u ) contains exactly two distinct characteristic periodic balanced sequences (not just one),which take the form ( w ∞ and ( w ∞ where w is the central word having the same slope as u .The following useful result is due to de Luca [11]; in particular, see [11, Remark 1 and Proposi-tion 9] and also [9, Lemma 5]. Proposition 21. Let w be a central word in { , } ∗ . If w = w w = w w where w and w are(central) words, then P al ( w 0) = ( w (+) = w w w and P al ( w 1) = ( w (+) = w w w . We end this section with a result (Corollary 23 below) that will be particularly useful in the proof ofour main combinatorial theorem. Let us first recall that a non-empty finite word v over an alphabet A is said to be a Lyndon word (resp. anti-Lyndon word ) if v is a primitive word that is lexicographicallyless (resp. lexicographically greater) than all of its circular shifts with respect to a given total orderon A . Proposition 22. [3, Theorem 3.2 and Corollary 3.1] A non-empty finite word v ∈ { , } ∗ is a balancedLyndon word (resp. balanced anti-Lyndon word) with respect to the lexicographic order if and only if v = 0 w (resp. v = 1 w ) for some central word w ∈ { , } ∗ . As a direct consequence of the above proposition, we have the following result. Corollary 23. For any central word w ∈ { , } ∗ , the (primitive) words w and w are the lexico-graphically least and greatest words amongst their circular shifts. 4. P ROOF OF T HEOREM 13– Assertions (i) and (ii) are straightforward (see [16, Lemma 2.4]).In order to prove the other assertions, we first recall the inequalities that are equivalent to s = φ (0 u ) from Theorem 11:(1) u ≤ T k ( s ) ≤ u and T k ( s ) ≤ s for all k ≥ . – Assertion (iii) is a consequence of Gan’s result (Theorem 11) together with Theorem 9 (seeRemark 12).– We will now prove Assertions (iv) and (v). Suppose that u is not a characteristic Sturmian se-quence and that u contains both ’s and ’s. Then we know from Theorem 11 that s := φ (0 u ) isa periodic balanced sequence satisfying the inequalities in (1). Indeed, s cannot be Sturmian, forotherwise u would be a characteristic Sturmian sequence by Remark 12. By Remark 20, O ( s ) contains exactly two distinct characteristic periodic balanced sequences, givenby s := ( w ∞ and s := ( w ∞ where w ∈ { , } ∗ is the central word with the same slope as s . We now deduce from Corollary 23that the lexicographically least sequence in O ( s ) is (0 w ∞ = 0( w ∞ = 0 s and the lexicographically greatest sequence in O ( s ) is (1 w ∞ = 1( w ∞ = 1 s . Hence,(2) s ≤ T k ( s ) ≤ s for all k ≥ . Moreover, since s is the lexicographically greatest sequence in its shift-orbit (by the second inequalityin (1)), we have s = (1 w ∞ = 1 s .We will now show that s ≤ u ≤ s . Since s and s are the lexicographically least andgreatest elements in O ( s ) , the inequalities in (1) imply that u ≤ s and s ≤ u . Hence s ≤ u ≤ s ; that is, ( w ∞ ≤ u ≤ ( w ∞ .Furthermore, we note that there does not exist another central word z such that ( z ∞ ≤ u ≤ ( z ∞ . For if so, then the set [0 u , u ] := { s ∈ { , } N , u ≤ s ≤ u } would contain the peri-odic balanced sequences z ∞ = (0 z ∞ and z ∞ = (1 z ∞ , and hence all of the shiftsof the characteristic periodic balanced sequence ( z ∞ , since the former two sequences are thelexicographically least and greatest sequences in the shift-orbit of ( z ∞ (by Proposition 19 andCorollary 23). But by [16, Lemma 5.4], the set [0 u , u ] contains a unique periodic balanced shift-orbit. Therefore, since O (( w ∞ ) ⊆ [0 u , u ] , we must have z = w . We have thus established thefollowing lemma. Lemma 24. Suppose u is a sequence in { , } N \ { ∞ , ∞ } that is not characteristic Sturmian.Then there exists a unique central word w ∈ { , } ∗ such that ( w ∞ ≤ u ≤ ( w ∞ . Moreover, φ (0 u ) = (1 w ∞ . Assertions (iv) and (v) are direct consequences of the above lemma, and the last statement in thetheorem follows from Theorem 11. (cid:3) 5. A COROLLARY By Theorem 15, the set of all characteristic Sturmian sequences in { , } N is given by S = { u ∈ { , } N , ∃ v ∈ { , } N \ ( { , } ∗ ∞ ∪ { , } ∗ ∞ ) , u = P al ( v ) } . And it follows from Proposition 17 that the set of all characteristic periodic balanced sequences in { , } N is given by P = { ∞ , ∞ } ∪ P ∪ P ISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD 11 where P := { u ∈ { , } N , ∃ v ∈ { , } ∗ , u = ( P al ( v )01) ∞ } and P := { u ∈ { , } N , ∃ v ∈ { , } ∗ , u = ( P al ( v )10) ∞ } . Given a characteristic periodic balanced sequence in { , } N of the form s := ( P al ( v ) xy ) ∞ where v ∈ { , } ∗ and { x, y } = { , } , we let ¯ s denote the other characteristic periodic balanced sequencein the shift-orbit of s , i.e., ¯ s := ( P al ( v ) yx ) ∞ (see Remark 20).As an immediate consequence of Theorem 13, we obtain the following description of the lexico-graphic world. Corollary 25. We have L = { (01 ∞ , ∞ ) } ∪ L ∪ L ∪ L ∪ L ∪ L ∗ where L := { (0 ∞ , v ) , v ∈ { , } N } , L := { (1 u , ∞ ) , u ∈ { , } N } , L := { (0 u , v ) ∈ S ∪ P ) × { , } N , v ≥ u } , L := { (0 u , v ) ∈ P × { , } N , v ≥ u } , L ∗ := { (0 u , v ) ∈ ( { , } N \ S ∪ P )) × { , } N , ∃ s ∈ P , s ≤ u ≤ ¯ s , v ≥ s } . 6. H OW TO DETERMINE φ (0 u ) FOR A GENERIC SEQUENCE u The following theorem provides a method for determining the central word w such that φ (0 u ) =(1 w ∞ for any “generic” sequence u ∈ { , } N falling into Case (v) of Theorem 13. Hereafter, aprefix of u that is a central word is called a central prefix of u . Theorem 26. Suppose u is a sequence in { , } N \{ ∞ , ∞ } that is neither a characteristic Sturmiansequence nor a characteristic periodic balanced sequence. Let v be the longest central prefix of u .Then v is finite ( v = ε ), and φ (0 u ) is determined as follows. (i) If v = 1 k for some k ≥ , then φ (0 u ) = (1 k ∞ = (1 w p,q ∞ where p = k and q = k + 1 . (ii) If v = 0 k for some k ≥ , then φ (0 u ) = (10 k ) ∞ = (1 w p,q ∞ where p = 1 and q = k + 1 . (iii) Suppose v contains both ’s and ’s. Let v , v be the unique pair of central words such that v = v v = v v where ℓ := | v | + 2 and ℓ := | v | + 2 are coprime periods of v .Consider the prefix of length | v | + 4 of u , namely the prefix vxyz where x, y ∈ { , } and | z | = | v | + 2 . (a) If either xy = 01 and z > v , or xy = 10 and z < v , then φ (0 u ) = (1 v ∞ =(1 w p,q ∞ where p = | v | + 1 and q = | v | + 2 = ℓ + ℓ . (b) If either xy = 01 and z < v , or xy = 00 , then φ (0 u ) = (1 v ∞ = (1 w p,q ∞ where p = | v | + 1 and q = ℓ . (c) If either xy = 10 and z > v , or xy = 11 , then φ (0 u ) = (1 v ∞ = (1 w p,q ∞ where p = | v | + 1 and q = ℓ .Note. In Assertion (iii), it cannot happen that z = vxy when x = y . For instance, if xy = 01 and z = v , then u would begin with the following word: v v 01 = v v v v where the prefix v v v = v v is a central word, by Propositions 17 and 21. But then u has acentral prefix longer than v ; thus z = v . Similarly, if xy = 10 , then z = v .The following lemma is needed for the proof of Theorem 26. Lemma 27. Suppose v is a central word in { , } ∗ \ (0 ∗ ∪ ∗ ) . Let v , v be the unique pair of centralwords such that v satisfies the equation v = v v = v v . Then v v (resp. v v ) is a prefixof the characteristic periodic balanced sequence ( v ∞ (resp. ( v ∞ ).Note. By Propositions 17 and 21, the words v v and v v are central words since v v = P al ( v and v v = P al ( v . Proof of Lemma 27. We will prove only that ( v ∞ begins with the central word P al ( v 0) = v v ,since the proof of the other case is very similar.By Proposition 17, ℓ := | v | +2 and ℓ := | v | +2 are coprime periods of v where | v | = ℓ + ℓ − .In particular, since ℓ = | v | is a period of v with ℓ < | v | , there exists an integer k ≥ such that v = ( v k v ′ where v ′ is a (possibly empty) prefix of v , in which case v = ( v k − v ′ since v = v v . Moreover, since v is a palindrome, ˜ v ′ is a prefix of v , and therefore v ′ = ˜ v ′ , i.e., v ′ is a palindrome. Furthermore, v ′ is a prefix of v since its reversal v ′ is a suffix of v . We willnow show that the central word P al ( v 0) = v v is a prefix of the characteristic periodic balancedsequence ( v ∞ by considering five different cases according to the length of the palindrome v ′ .(1) If v ′ = v , then v = ( v k +1 and v = ( v k . Hence, v being a palindrome impliesthat v is a palindrome and we have v = ( v k +1 = (01 v ) k +1 . Therefore, ( v ∞ = ( v k +1 v |{z} v ( v ∞ = v v ( v ∞ . Thus, the central word P al ( v 0) = v v is a prefix of ( v ∞ .(2) If v ′ = v , then since v ′ and v are palindromes, we have v v , and hence v is apower of ; in particular, v = 1 ℓ − . Therefore ( v ∞ = ( v k v v v ∞ = ( v k v | {z } v ℓ − | {z } v v ∞ = v v v ∞ = P al ( v v ∞ . (3) If v ′ = v , then v = ( v k v , and therefore v = ( v k − v . But this implies that ℓ = kℓ , which is impossible since ℓ and ℓ are coprime integers greater than . ISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD 13 (4) If v = v ′ , then since v and v ′ are palindromes, we have v ′ v ′ . Therefore v ′ (andhence v ) is a power of ; in particular, v = 0 ℓ − . Thus ( v ∞ = ( v k v v v ∞ = ( v k v ′ | {z } v v |{z} v v ∞ = v v v ∞ = P al ( v v ∞ . Note that we cannot have v = v ′ because v ′ and v are both prefixes of v .(5) If | v ′ | ≤ | v | − , then v = v ′ v ′′ for some (possibly empty) word v ′′ ∈ { , } ∗ , in whichcase v = ( v ′ v ′′ k v ′ . (Note that neither v ′ nor v ′ is a prefix of v because v ′ and v are both prefixes of v .) Since v is a palindrome that begins with the palindrome v = v ′ v ′′ and therefore ends with ˜ v = v = ˜ v ′′ v ′ , we see that ˜ v ′′ v ′ = v ′′ v ′ . Hence v ′′ isa palindrome. Moreover, v ′′ is a central word since v ′′ is a palindromic prefix (and also apalindromic suffix) of the central word v and any palindromic prefix (or suffix) of a centralword is central (see [12] or [19, Corollary 2.2.10]). Thus, by Proposition 17, v satisfies theequation v = v ′′ v ′ = v ′ v ′′ . Hence, we have ( v ∞ = ( v k v ′ v ′′ | {z } v v v ∞ = ( v k v ′ v ′′ v ′ | {z } v v ′′ v ∞ = v v v ′′ v ∞ = P al ( v v ′′ v ∞ . In all of the above cases (with the exception of the impossible case (3)), we have shown that thecentral word P al ( v 0) = v v is a prefix of ( v ∞ , as required. (cid:3) Proof of Theorem 26. Suppose u is a sequence in { , } N \ { ∞ , ∞ } that is neither a characteristicSturmian sequence nor a characteristic periodic balanced sequence. Then the longest central prefix of u , say v , is non-empty since it could (at the very least) be a letter. Furthermore, v is finite; otherwise, if v were infinite, then u would be either a characteristic Sturmian sequence or a characteristic periodicbalanced sequence (see Theorem 15 and Remark 18).We know from Theorem 13 (or Lemma 24) that there exists a unique central word w ∈ { , } ∗ such that ( w ∞ < u < ( w ∞ , in which case φ (0 u ) is equal to the periodic balanced sequence (1 w ∞ . We will show how to determine w in terms of the longest central prefix v . Note that w iseither empty or a (palindromic) prefix of v , by the maximality of v .First suppose that v = x k for some x ∈ { , } and k ≥ . Then by the maximality of v as a centralprefix of u , it follows that u begins with x k y = vy where y ∈ { , } , y = x . Moreover, the prefix oflength k + 1 of u takes the form x k yu where | u | = k and | u | x ≤ k − ; otherwise u would beginwith x k yx k = P al ( x k y ) , contradicting the fact that v (= P al ( x k )) is the longest central prefix of u . If x = 1 , then we easily see that (1 k − ∞ < u (= 1 k u · · · ) < (1 k ∞ , where the latter inequality follows from the fact that | u | = k and u < k (since u contains at most k − occurrences of the letter ). Hence by Lemma 24, φ (0 u ) = (1 k ∞ = (1 w p,q ∞ where p = k and q = k + 1 . Similarly, if x = 0 , we have (0 k ∞ < u (= 0 k u · · · ) < (0 k − ∞ , where the first inequality follows from the fact that | u | = k and k < u (since u contains at most k − occurrences of the letter ). Hence by Lemma 24, φ (0 u ) = (10 k ) ∞ = (1 w p,q ∞ where p = 1 and q = k + 1 . We have thus proved Assertions (i) and (ii) of the theorem.Now suppose that the longest central prefix v of u contains both 0’s and 1’s. Then by Proposition 17,there exists a unique pair of central words v , v ∈ { , } ∗ such that v = v v = v v where ℓ := | v | + 2 and ℓ := | v | + 2 are coprime periods of v , and min { ℓ , ℓ } is the minimal period of v .Consider the prefix of length | v | + 4 of u , namely the prefix vxyz where x, y ∈ { , } and | z | = | v | + 2 . We will now prove each of the cases (a), (b), and (c) of Assertion (iii). Case (a): Let us first suppose that u begins with v z where | z | = | v | and z > v . Then itis easy to see that ( v ∞ < u < ( v ∞ . Hence, by Lemma 24, we have φ (0 u ) = (1 v ∞ . Moreover, v = w p,q where p = | v | + 1 and q = | v | + 2 = ℓ + ℓ . Similarly, if u begins with v z where | z | = | v | and z < v ,then ( v ∞ < u < ( v ∞ , and therefore φ (0 u ) = (1 v ∞ by Lemma 24. Case (b): In this case, either u begins with v or u begins with v z where | z | = | v | and z < v .Since v is a prefix of v (which in turn is a prefix of u ), we have ( v ∞ < u . Fur-thermore, by Lemma 27, the characteristic periodic balanced sequence ( v ∞ begins withthe central word P al ( v 0) = v v . Thus, if u begins with v , then u < ( v ∞ . Onthe other hand, if u begins with v z where | z | = | v | and z < v , then we will showthat u < ( v ∞ by considering the prefix of length | v | + ℓ of u , namely v z where | z | = | v | . We first note that z ≤ v since v is a prefix of v and z is a prefix of z where z and v satisfy z < v . Furthermore, z < v (i.e., z = v ). Otherwise, if z = v , then u would begin with the central word P al ( v 0) = v v . But then u would have a centralprefix that is longer than v ; a contradiction. Therefore z < v , and hence u < ( v ∞ since ( v ∞ begins with v v where v > z , as shown above. Case (c): This case is symmetric to Case (b). (cid:3) Example 28. The following examples demonstrate the computation of φ (0 u ) for sequences u in { , } N that are neither characteristic Sturmian nor periodic and balanced. Where appropriate, thelongest central prefix of the sequence is highlighted in boldface. ISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD 15 (1) The following two general facts can easily be deduced from the proofs of parts (i) and (ii) ofTheorem 26.(a) φ (0 u ) = (1 k ∞ for any sequence u having a prefix of the form k v where k ≥ , | v | = k , and | v | ≤ k − .(b) φ (0 u ) = (10 k ) ∞ for any sequence u having a prefix of the form k v where k ≥ , | v | = k , | v | ≤ k − .(2) By part (iii)(a) of Theorem 26, φ (0 u ) = (10100100) ∞ = (1 w , ∞ for any sequence u beginning with P al (010)011 = w , 011 = . (3) Let u be the (non-characteristic) Sturmian sequence f = · · · where f is the (binary) Fibonacci sequence (see Examples 2 and 16). Then the longest centralprefix of u is w , = 101 = P al (10) and u begins with w , . Therefore, by part (iii)(b) ofTheorem 26, we have φ (0 u ) = φ (01 f ) = (10) ∞ = (1 w , ∞ .(4) By part (iii)(c) of Theorem 26, φ (0 u ) = (10100) ∞ = (1 w , ∞ for any sequence u begin-ning with P al (010)101 = w , 101 = . (5) By parts (iii)(b) and (iii)(c) of Theorem 26, φ ( x ) = (10) ∞ = (1 w , ∞ for any sequence x beginning with or . In particular, φ (0 t ) = (10) ∞ for the Thue-Morse sequence t ,which is the fixed point beginning with of the morphism , : t = · · · Also note that φ ( t ) = (110) ∞ = (1 w , ∞ .(6) Recall that the central word w p,q of slope p/q ∈ (0 , (where gcd( p, q ) = 1 ) has length q − and contains p − occurrences of (and q − p − occurrences of ). We observe that if p > q (i.e., if w p,q contains more ’s than ’s), then w p,q begins with ; otherwise, if p < q ,then w p,q begins with . Hence, we deduce the following general facts from part (iii)(a) ofTheorem 26.(a) If p > q , then φ (0 u ) = (1 w p,q ∞ for any sequence u beginning with w p,q .(b) If p < q , then φ (0 u ) = (1 w p,q ∞ for any sequence u beginning with w p,q . Remark . To determine the longest central prefix of a sequence u ∈ { , } N (which is neither acharacteristic Sturmian sequence nor a characteristic periodic balanced sequence), possibly the easiestway is to check each palindromic prefix of u (in order of increasing length) to see if it is equal to P al ( u ) for some u ∈ { , } ∗ , until there are no more palindromic prefixes or until one reaches apalindromic prefix that is not in the image of P al . Note. Theorem 26 also provides a method for computing F ( x ) in Case (v) of Theorem 7. For example, F ( ) = since the base 2 expansion of is · · · (or · · · ) and we have φ (01000 · · · ) =(10) ∞ = φ (00111 · · · ) where (10) ∞ is the base 2 expansion of / . (See part (1) of Example 28above.) 7. 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Veerman, Symbolic dynamics of order-preserving orbits, Physica D (1987), 191–201.CNRS, LRI, UMR 8623, U NIVERSIT ´ E P ARIS -S UD , B ˆ ATIMENT RSAY C EDEX , FRANCE E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS AND S TATISTICS , S CHOOL OF C HEMICAL AND M ATHEMATICAL S CIENCES ,M URDOCH U NIVERSITY , P ERTH , WA 6150, AUSTRALIA Current address : The Mathematics Institute, Reykjav´ık University, Kringlan 1, IS-103 Reykjav´ık, ICELAND E-mail address ::