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SUBWORD COMPLEXITY AND LAURENT SERIES WITH COEFFICIENTS IN A FINITE FIELD 0 1 ALINAFIRICEL 0 2 n a J Abstract. Decimalexpansionsofclassicalconstantssuchas√2,πand ζ(3)havelongbeenasourceofdifficultquestions. InthecaseofLaurent 4 1 series with coefficients in a finite field, where no carry-over difficulties appear, the situation seems to be simplified and drastically different. ] On the other hand, Carlitz introduced analogs of real numbers such as T π,eorζ(3). Hence,itbecamereasonabletoenquirehow“complex” the N Laurent representation of these “numbers” is. h. In this paper we prove that the inverse of Carlitz’s analog of π, Πq, has in general a linear complexity, except in the case q = 2, when the t a complexityisquadratic. Inparticular,thisimpliesthetranscendenceof m Π2 over F2(T). In the second part, we consider the classes of Laurent [ series of at most polynomial complexity and of zero entropy. We show that these satisfy some niceclosure properties. 1 v 8 1. Introduction and motivations 4 5 A long standing open question concerns the digits of the real number 2 . π = 3.14159 . The decimal expansion of π has been calculated to billions 1 ··· of digits and unfortunately, there are no evident patterns occurring. Actu- 0 0 ally,foranyb 2, theb-aryexpansionofπ lookslikearandomsequence(see ≥ 1 for instance [10]). More concretely, it is widely believed that π is normal, : v meaning that all blocks of digits of equal length occur in the b-ary represen- i tation of π with the same frequency, but current knowledge on this point is X scarce. r a Ausualwaytodescribethedisorderofaninfinitesequencea = a a a 0 1 2 ··· is to compute its subword complexity, which is the function that associates to each positive integer m the number p(a,m) of distinct blocks of length m occurring in the word a. Let α be a real number and let a be the representation of α in an integral base b 2. The complexity function of α is defined ≥ as follows: p(α,b,m) = p(a,m), for any positive integer m. Notice that π being normal would imply that its complexity must be maximal, that is p(π,b,m) = bm. In this direction, similar questions have been asked about other well-known constants like e, log2, ζ(3) or √2 and it is widely believed that the following conjecture is true. 1 2 ALINAFIRICEL Conjecture 1.1. Let α be one of the classical constants: π, e, log2, ζ(3) and √2. The complexity of the real number α satisfies: p(α,b,m) =bm, for every positive integer m and every base b 2. ≥ We mention that in all this paper we will use Landau’s notations. We write f(m) = Θ(g(m)) if there exist positive real numbers k ,k ,n such 1 2 0 that, for every n > n we have 0 k g(n) < f(n) < k g(n) . 1 2 | | | | | | We write also f(m)= O(g(m)) if there exist two positive real numbers k,n 0 such that, for every n n we have: 0 ≥ f(n) < k g(n) . | | | | If α is a rational real number then p(α,b,m) = O(1), for every integer b 2. Moreover, there is a classical theorem of Morse and Hedlund [28] ≥ which states that an infinite sequence a = (a ) is eventually periodic if n n≥0 and only if p(a,m) is bounded. If not, the complexity function is strictly increasing. In particular, (1) p(a,m) m+1, ≥ for every nonnegative integer m. A sequence which saturates the inequality above is called a Sturmian sequence (see the original papers of Morse and Hedlund [28, 29]). According to this theorem, an irrational real number α has a complexity function which satisfies p(α,b,m) m+1, for every m N. Concerning ≥ ∈ irrational algebraic numbers, the main result known to date in this direction is due to Adamczewski and Bugeaud [3]. These authors proved that the complexity of an irrational algebraic real number α satisfies p(α,b,m) lim = + , m→∞ m ∞ for any base b 2. ≥ For more details about complexity of algebraic real numbers, see [3, 4]. For classical transcendental constants, there is a more ambiguous situation and, to the best of our knowledge, the only result that improves the bound following from Inequality (1) was recently proved in [1]. It concerns the real number eand someother exponential periods. More precisely, Adamczewski showedthatifξ isanirrationalrealnumberwhoseirrationalexponentµ(ξ) = 2, then lim p(ξ,b,m) m = + , m→∞ − ∞ for any base b 2. ≥ The present paper is motivated by this type of questions, but asked for Laurent series with coefficients in a finite field. In the sequel we will denote SUBWORD COMPLEXITY AND LAURENT SERIES 3 respectivelybyF (T),F [[T−1]]andF ((T−1))thefieldofrationalfunctions, q q q the ring of formal series and the field of Laurent series over the finite field F , q being a power of a prime number p. q Let us also recall the well-known analogy between integers, rationals and real numbers on one side, and polynomials, rationals functions, and Laurent series with coefficients in a finite field, on the other side. Notice that, the coefficients in F play the role of “digits” in the basis given by the powers q of the indeterminate T. There is still a main difference: in the case of real numbers, it is hard to control carry-overs when we add or multiply whereas in the case of power series over a finite field, this difficulty disappear. By analogy with the real numbers, the complexity of a Laurent series is defined as the subword complexity of its sequence of coefficients. Again, the theorem of Morse and Hedlund gives a complete description of the rational Laurent series; more precisely, they are the Laurent series of bounded complexity. Hence, most interesting questions concern irrational series. There is a remarkable theorem of Christol [18] which describes precisely thealgebraicLaurentseriesoverF (T)asfollows. Letf(T)= a T−n q n≥−n0 n be a Laurent series with coefficients in F . Then f is algebraic over F (T) q q P if, and only if, the sequence of coefficients (a ) is p-automatic. n n≥0 For more references on automatic sequences, see for example [8]. Further- more,Cobhamprovedthatthesubwordcomplexityofanautomaticsequence is at most linear [20]. Hence, an easy consequence of those two results is the following. Theorem 1.1. Let f F ((T−1)) algebraic over F (T). Then we have: q q ∈ p(f,m)= O(m). The reciprocal is obviously not true, since there are uncountable many Laurent series with linear complexity. In contrast with real numbers, the situation is thus clarified in the case of algebraic Laurent series. Also, notice that Conjecture 1.1 and Theorem 1.1 point out the fact that the situations in F ((T−1)) and in R appear to be completely opposite. q On the other hand, Carlitz introduced [21] functions in positive characteristic by analogy with the number π, the Riemann ζ function, the usual exponential and the logarithm function. Many of these were shown to be transcendental over F (T) (see [22, 26, 33, 34, 35]). In the present paper we q focus on the analog of π, denoted, for each q, by Π , and we prove that its q inverse has a “low” complexity. Moreprecisely, we will prove inSection 3 the following results. Theorem 1.2. Let q = 2. The complexity of the inverse of Π satisfies: q 1 p ,m = Θ(m2). Π (cid:18) 2 (cid:19) 4 ALINAFIRICEL Theorem 1.3. Let q 3. The complexity of the inverse of Π satisfies: q ≥ 1 p ,m = Θ(m). Π q (cid:18) (cid:19) Since any algebraic series has a linear complexity (by Theorem 1.1), the following corollary yields. Corollary 1.1. Π is transcendental over F (T). 2 2 The transcendence of Π over F (T) was first proved by Wade in 1941 q q (see [34]) using an analog of a classical method of transcendence in zero characteristic. Another proof was given by Yu in 1991 (see [35]), using the theory of Drinfeld modules. Then, de Mathan and Cherif, in 1993 (see [22]), using tools from Diophantine approximation, proved a more general result, but in particular their result implied the transcendence of Π . q Christol’s theorem has also been used as a combinatorial criterion in order to prove the transcendence of Π . This is what is usually called an q “automatic proof”. The non-automaticity and also the transcendence, was first obtained by Allouche, in [6], via the so-called q-kernel. Notice that our proof of transcendence here is based also by Christol’s theorem, but we obtain the non-automaticity of Π over F (T) as a consequence of the subword 2 2 complexity. Furthermore, motivated by Theorems 1.2, 1.3 and by Conjecture 1.1, we consider the classes of Laurent series of at most polynomial complexity P and of zero entropy (see Section 4), which seem to be good candidates Z to enjoy some nice closure properties. In particular, we prove the following theorem. Theorem 1.4. and are vector spaces over F (T). q P Z Another motivation of this work is the article [11] of Beals and Thakur. These authors proposed a classification of Laurent series in function of their space or time complexity. This complexity is in fact a characteristic of the (Turing) machine that computes the coefficient a , if f(T) := a T−i. i i i They showed that someclasses of Laurent series have good algebraic proper- P ties (for instance, the class of Laurent series corresponding to any determin- istic space class at least linear form a field). They also place some Carlitz’s analogs in the computational hierarchy. This paper is organized as follows. Some definitions and basic notions on combinatorics on words and Laurent series are recalled in Section 2. Section 3 is devoted to the study of the Carlitz’s analog of π; we prove Theorems 1.2 and 1.3. In Section 4 we study some closure properties of Laurent series of “low” complexity (addition, Hadamard product, derivative, Cartier oper- ator) and we prove Theorem 1.4; in particular, this provides a criterion of linear independence over F (T) for two Laurent series in function of their q SUBWORD COMPLEXITY AND LAURENT SERIES 5 complexity. Finally, we conclude in Section 5 with some remarks concerning the complexity of the Cauchy product of two Laurent series, which seems to be a more difficult problem. 2. Terminologies and basic notions In this section, we briefly recall some definitions and well-known results from combinatorics on words. Moreover, we recall some basic notions on algebraic Laurent series. A word is a finite, as well as infinite, sequence of symbols (or letters) be- longing to a nonempty set , called alphabet. We usually denote words by A juxtaposition of theirs symbols. Given an alphabet , we denote by ∗ := ∞ k the set of finite words A A ∪k=0A over . Let V := a a a ∗. Then the integer m is the length of V 0 1 m−1 A ··· ∈ A and is denoted by V . The word of length 0 is the empty word, usually de- | | noted by ε. We also denote by m the setof all finite words of length m and by N the set of all infinite woArds over . We typically use the uppercase A A italic letters X,Y,Z,U,V,W to represent elements of ∗. We also use bold lowercase letters a,b,c,d,e,f to represent infinite woArds. The elements of are usually denoted by lowercase letters a,b,c, . A ··· We say that V is a factor (or subword) of a finite word U if there exist some finite words A, B, possibly empty such that U = AVB and we denote it by V ⊳U. Otherwise, V ⋪ U. We say that X is a prefix of U, and we denote by X U if there exists Y such that U = XY. We say that Y is a p ≺ suffix of U, and we denote by Y U if there exists X such that U = XY. s ≺ Also, we say that a finite word V is a factor (or subword) of an infinite word a = (a ) if there exists a nonnegative integer j such that n n≥0 V = a a a . The integer j is called an occurence of V. j j+1 j+m−1 ··· Let U,V,W be three finite words over , V possibly empty. We denote: A i(U,V,W) := AVB, A U, B W, A,B possibly empty , s p { ≺ ≺ } and i(U,V,W)+ := AVB, A U, B W, A,B nonempty . s p { ≺ ≺ } If n is a nonnegative integer, we denote by Un := UU U. We denote ··· ntimes also U∞ := UU , that is U concatenated (with itself) infinitely many ··· | {z } times. An infinite word a is periodic if there exists a finite word V such that a = V∞. An infinite word is eventually periodic if there exist two finite words U and V such that a =UV∞. 6 ALINAFIRICEL The fundamental operation on words is concatenation. Notice that ∗, A together with concatenation, form the free monoid over , whose neutral A element is the empty word ε. 2.1. Subword complexity. Let a be an infinite word over . As already A mentioned in Introduction, the subword complexity of a is the function that associates to each m N the number p(a,m) defined as follows: ∈ p(a,m) = Card (a ,a ,...,a ), j N . j j+1 j+m−1 { ∈ } For any word a, p(a,0) = 1 since, by convention, the unique word of length 0 is the empty word ε. For example, let us consider the infinite word a = aaa , the concate- ··· nation of a letter a infinitely many times. It is obvious that p(a,m) = 1 for any m N. More generally, if a is eventually periodic, then its complexity ∈ function is bounded. On the other side, let us consider the infinite word of Champernowne over the alphabet 0,1,2,3,...,9 , a:= 0123456789101112 . Notice that { } ··· p(a,m) = 10m for every positive integer m. More generally, one can easily prove that for every m N and for every ∈ word a over the alphabet , we have the following: A 1 p(a,m) (card )m. ≤ ≤ A We give now an important tool we shall use in general, in order to obtain aboundof thesubword complexity function(foraproofseeforexample [8]): Lemma 2.1. Let a be an infinite word over an alphabet . We have the A following properties: p(a,m) p(a,m+1) card p(a,m), for every integer m 0; • p(a,m+≤n) p(a,m)p≤(a,n),Afo·r all integers m,n 0. ≥ • ≤ ≥ Let F be the finite field with q elements, where q is a power of a prime q number p. In this paper, we are interested in Laurent series with coefficients in F . q Let n N and consider the Laurent series: 0 ∈ +∞ f(T)= a T−n F ((T−1)). n q ∈ n=X−n0 Letmbeanonnegative integer. Wedefine the complexity off,denotedby p(f,m), as being equal to the complexity of the infinite word a = (a ) . n n≥0 SUBWORD COMPLEXITY AND LAURENT SERIES 7 2.2. Topological entropy. Let a be an infinite word over an alphabet . A The (topological) entropy of a is defined as follows: logp(a,m) h(a) = lim . m→∞ m Thelimit existsasaneasyconsequence of thefollowing property: p(a,n+ m) p(a,n)p(a,m), for every m,n 0 (which is the second part of the ≤ ≥ Lemma 2.1). If the base of the logarithm is the cardinality of the alphabet then: 0 h(a) 1. ≤ ≤ Notice that, by definition, the “simpler” the sequence is, the smaller its entropy is. Let n N and consider the Laurent series 0 ∈ +∞ f(T)= a T−n F ((T−1)). n q ∈ n=X−n0 We define the entropy of f, denoted by h(f), as being equal to the entropy of the infinite word a = (a ) . n n≥0 2.3. Morphisms. Let (respectively ) be an alphabet and let ∗ (re- A B A spectively ∗) be the corresponding free monoid. A morphism σ is a map B from ∗ to ∗ such that σ(UV) =σ(U)σ(V) for all words U,V ∗. Since A B ∈ A the concatenation is preserved, it is then possible to define a morphism on . A If = we can iterate the application of σ. Hence, if a , σ0(a) = a, A B ∈ A σi(a) = σ(σi−1(a)), for every i 1. ≥ Letσ : beamorphism. Theset ∗ N isendowed withanatural A → A A ∪A topology. Roughly, two words are close if they have a long common prefix. We can thus extend the action of a morphism by continuity to ∗ N. A ∪ A Then, a word a N is a fixed point of a morphism σ if σ(a) = a. ∈ A A morphism σ is prolongable on a if σ(a) = ax, for some x + := ∈ A ∈ A ∗ ε . If σ is prolongable then the sequence (σi(a)) converges to the i≥0 A \{ } infinite word: σ∞(a) = lim σi(a) = axσ(x)σ2(x)σ3(x) . i→∞ ··· Example 2.1. The Fibonacci word f = 0100101001001 is an example ··· of an infinite word generated by iterating the morphism: σ(0) = 01 and σ(1) = 0. More precisely, f = σ∞(0) is the unique fixed point of σ. The order of growth of a letter x is the function σn(x) , for n 0. In | | ≥ general, this function is bounded or, if not, is growing asymptotically like the function naxbn. A morphism is said to be polynomially diverging if there x exists b > 1 such that, for any letter x, the order of growth of x is naxbn and a 1 for some x. A morphism is exponentially diverging if every letter x x ≥ 8 ALINAFIRICEL has the order of growth naxbn with b > 1 and not all b are equal. For more x x x details the reader may refer to [30]. A morphism σ is said to be uniform of length m 2 if g(x) = m. Notice ≥ | | that a word generated by an uniform morphism of length m is m-automatic (see for example [20]). In particular, its complexity is O(1) if the word is eventually periodic; otherwise, it is Θ(m). For moreabout thecomplexity function ofwords generated by morphisms there is a classical theorem of Pansiot [30] that characterizes the asymptotic behavior of factor complexity of words obtained by iterating a morphism. 2.4. AlgebraicLaurentseries. ALaurentseriesf(T)= a T−n n≥−n0 n ∈ F ((T−1)) is said to be algebraic over the field F (T) if there exist an integer q q P d 1 and polynomials A (T),A (T),...,A (T), with coefficients in F and 0 1 d q ≥ not all zero, such that: A +A f + +A fd = 0. 0 1 d ··· Otherwise, f is transcendental over F (T). q Let us now give an example of Laurent series algebraic over the field of rational functions. Example 2.2. Let us consider the formal series f(T) = c T−n n≥0 n ∈ F [[T−1]] where c:= (c ) is the Cantor sequence defined as follows: 3 n n≥0 P 1 if (n) contains only 0 and 2; 3 c = . n (0 if (n)3 contains the letter 1. Here (n) denotes the expansion in base 3 of n. By definition, we get that 3 c = c = c and c = 0, for all n N. 3n n 3n+2 3n+1 ∈ We have: f(T) = c T−3n+ c T−3n−1+ c T−3n−2 3n 3n+1 3n+2 n≥0 n≥0 n≥0 X X X = c T−3n+ c T−3n−2. n n n≥0 n≥0 X X Hence, f(T)= f(T3)+T−2f(T3) and, since we are in characteristic 3, we obtain that f satisfies the following equation: (1+T2)f2(T) T2 = 0. − Thus f is an algebraic Laurent series. Notice also that, the infinite sequence c is 3-automatic, as predicted by Christol’s theorem, and in particular the complexity of c satisfies: p(c,m) = O(m). SUBWORD COMPLEXITY AND LAURENT SERIES 9 3. An analogue of Π In1935,Carlitz[21]introducedforfunctionfieldsinpositivecharacteristic an analog of the exponential function defined over , which is the comple- ∞ C tion of the algebraic closure of F ((T−1)) (this is the natural analogue of the q complex numbers field). In order to get good properties in parallel with the complex exponential, the resulting analogue, z e (z), satisfies: C → e (0) = 0, d/dz(e (z)) = 1 and e (Tz) = Te (z)+e (z)q. C C C C C This is what we call the Carlitz exponential and the action u Tu + → uq leads to the definition of the Carlitz F [T]-module, which is in fact a q particular case of Drinfeld module. The Carlitz exponential, e (z), may be C defined by the following infinite product: z e (z) = z (1 ) C − aΠ a∈FqY[T],a6=0 q where e ∞ −1 q 1 Πq = (−T)q−1 1− Tqj−1 . j=1(cid:18) (cid:19) Y Since ez = 1 if ande only if z 2πiZ and since e (z) was constructed by C ∈ analogy such that e (z) = 0 if and only if z Π F [T] (in other words the C q q ∈ kernel of e (z) is Π F [T]), we get a good analogue Π of 2πi. In order to C q q q obtain a good analogue of the real number π, wee take its one unit part and hence we obtain: e e ∞ −1 1 Π = 1 . q − Tqj−1 j=1(cid:18) (cid:19) Y For more details about analogs given by the theory of Carlitz modules, and in particular about the exponential function or its fundamental period Π , we refer the reader to the monographs [26, 33]. q If we look for the Laurent series expansion of Π , then we obtain that e q ∞ −1 1 Π = 1 = a T−n, q − Tqj−1 n j=1(cid:18) (cid:19) n≥0 Y X where a is defined as the number of partitions of n whose parts take values n in I = qj 1,j 1 , taken modulo p. { − ≥ } To compute the complexity of Π , we would like to find a closed formula q or some recurrence relations for the sequence of partitions (a ) . This n n≥0 question seems quite difficult and we are not able to solve it at this moment. However, itwasshownin[6]thattheinverseofΠ hasthefollowingsimple q Laurent series expansion: 10 ALINAFIRICEL ∞ ∞ 1 1 = 1 = p X−n Πq − Xqj−1 n j=1(cid:18) (cid:19) n=0 Y X where the sequence p = (p(n)) is defined as follows: q n≥0 (2) 1 if n = 0; p = ( 1)cardJ if there exists a set J N∗ such that n = (qj 1); n  − ⊂ j∈J − 0 if there is no set J N∗ such that n = (qj 1). ⊂ j∈J −P We mention that if such a decomposition exists, iPt is unique. In the rest of this section we will prove Theorem 1.2 and 1.3. 3.1. Proof of Theorem 1.2. In this part we study the sequence p = 2 (2) (p ) ,definedbytheformula(2)inthecasewhere q = 2. Moreprecisely: n n≥0 (3) 1 if n = 0 or if there exists J N∗ such that n = (2j 1); p(2) = ⊂ j∈J − n (0 otherwise. P In order to lighten the notations, in the rest of this subsection we set (2) p := p so that p = p p p . n n 2 0 1 2 ··· For every n 1, we denote by W the factor of p that occurs between n 2 ≥ positions 2n 1 and 2n+1 2, that is: − − Wn := p2n−1 p2n+1−2. ··· We also set W := 1. Observe that W = 2n. 0 n | | With these notations the infinite word p can be factorized as: 2 p = 1 10 110011011000 = W W W . 2 0 1 2 ··· ··· W0 W1 W2 W3 In order to compute the complexity function of p , we need the following |{z}|{z}|{z}| {z } 2 lemmas. Lemma 3.1. Let n and k be two positive integers such that: k < 2n 1. − Then k can be written as (2j 1) if and only if k +(2n 1) can be j∈J − − written as (2i 1), where I and J are finite subsets of N∗. i∈I − P Remark 3P.1. Thisisequivalenttosaythatak = 1ifandonlyifak+(2n−1) = 1. Proof. The first part is pretty obvious. If k = (2j 1), then: j∈J − k+(2n 1) = (2j 1)+(2n 1)P= (2j 1). − − − − j∈J j∈J∪{n} X X For the second part, let supposethat k+(2n 1) = (2i 1). We prove − i∈I − that k can be also represented in this form. More precisely, we show that n I and consequently k = (2i 1). P ∈ i∈I\{n} − P