On possible growths of arithmetical complexity

DOI: 10.1051/ita:2006021

ON POSSIBLE GROWTHS OF ARITHMETICAL COMPLEXITY

Anna E. Frid

Abstract. The arithmetical complexity of infinite words, defined by Avgustinovich, Fon-Der-Flaass and the author in 2000, is the number of words of lengthnwhich occur in the arithmetical subsequences of the infinite word. This is one of the modifications of the classical function of subword complexity, which is equal to the number of factors of the infi- nite word of lengthn. In this paper, we show that the orders of growth of the arithmetical complexity can behave as many sub-polynomial functions. More precisely, for each sequenceuof subword complexity fu(n) and for each primep≥3 we build a Toeplitz word on the same alphabet whose arithmetical complexity isa(n) = Θ(nfu(log_pn)).

Mathematics Subject Classification.68R15, 68Q45.

Introduction

Arithmetical complexity of an infinite word is a function relative to its subword complexity. It is larger than or equal to the subword complexity since it counts not only factors of the word but all words occurring in its arithmetic progressions.

By the definitions, if the infinite word isu=u1u2· · ·u_k· · ·, whereu_iare symbols, then for allnits subword complexity f_u(n) is the number of distinct words of the formu_iu_i+1· · ·u_i+n−1for arbitraryi, and its arithmetical complexitya_u(n) is the number of all words of the formu_ku_k+d· · ·u_k+(n−1)d for arbitrary initial symbols kand differencesd. Note that by the classical Van der Waerden theorem [23, 25], there are arbitrarily large powers of some symbol among these words.

Keywords and phrases.Arithmetical complexity, infinite word, subword complexity, Toeplitz word, bispecial words.

∗ Supported in part by RFBR grants 06-01-00694 and 06-01-00694 and Russian Science Support Foundation.

1 Sobolev Institute of Mathematics SB RAS, Koptyug av., 4, 630090 Novosibirsk, Russia;

frid@math.nsc.ru

c EDP Sciences 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/itaor http://dx.doi.org/10.1051/ita:2006021

Arithmetical complexity was introduced in 2000 by Avgustinovich, Fon-Der- Flaass and Frid [5]; its idea belongs to S.V. Avgustinovich. It has become one of the most well-explored modified complexity functions relative to the subword com- plexity. Other definitions included-complexity introduced in 1987 by Iv´anyi [11];

palindrome complexity(see Alloucheet al. [1] for a survey); modified complexity by Nakashima, Tamura, and Yasutomi [21, 22]; pattern complexity introduced in 2002 by Restivo and Salemi [24]; andmaximal pattern complexityby Kamae and Zamboni [16] which is also dated 2002 and well-explored [16–19]. For a survey on subword complexity, see Ferenczi [12].

All published results on arithmetical complexity concern words of linear sub- word complexity. Arithmetical complexity of such words can behave variously: it can be exponential, as for the Thue-Morse word, or linear, as for the paperfolding word [5]. In subsequent papers, uniformly recurrent words of linear arithmetical complexity were characterized [15], and a family of words having lowest complex- ity was found among them [4]. On the other hand, the arithmetical complexity of Sturmian words has been estimated as Θ(n³) and found for Sturmian words of many slopes [8, 13].

Here we find a variety of rates of growth of arithmetical complexity and show that it is not less than the variety of possible subword complexity rates of growth:

each function of the subword complexity of a word can be included to a formula for the arithmetical complexity of another word. More precisely, recall that the notation f(n) = Θ(g(n)) for functions f and g of positive integers means that C1g(n)≤f(n)≤ C2g(n) for all sufficiently largenand some positive constants C1 andC2. We prove the following

Theorem 0.1. For each infinite word uon a finite alphabet Σ, of subword com- plexityf_u(n), and for each prime p≥3 there exists an infinite word on Σwhose arithmetical complexity isΘ(nf_u(log_pn)).

In the next section, we describe the (well-known) technique of bispecial words which is used below for estimates of arithmetical complexity. Then in Section 2 the construction of the required word is introduced; this word is a Toeplitz word of a special kind, it depends onu, and its subword complexity is always linear. In the next four sections, we analyse the set of arithmetical factors of the constructed word, find its bispecial words and relate them to factors ofu; compute their pa- rameters and at last obtain estimates for the arithmetical complexity, thus proving Theorem 0.1. After that in Section 7 we discuss particular cases of Theorem 0.1.

1. Complexities and bispecial words

Let us define notions of complexity of words and languages used in this paper and discuss ways to compute them.

Let Σ be a finite alphabet of cardinalityq; as usual, the set of all finite words on Σ including the empty wordλwill be denoted by Σ^∗, and the set of all (right) infinite words on Σ will be denoted by Σ^ω. A finite word v is called a factor, or subword, of a finite or infinite word uifu=s1vs2 for some (possibly empty)

Figure 1. A bispeciality graph.

wordss1 ands2. A languageF ⊆Σ^∗is calledfactorialif it is closed under taking factors. Clearly, the languageF(u) of factors of a finite or infinite worduis always factorial.

A wordvis called anarithmetical subwordof a wordu=u1u2· · ·u_m· · ·, where u_i ∈ Σ for all i, if v = u_ku_k+d· · ·u_k+(n−1)d for some k and d. The set of all arithmetical subwords ofu(or of words from a languageF) is calledarithmetical closureof u (or F) and denoted byA(u) (A(F)). The term “closure” is chosen sinceA(A(F)) =A(F) for all F; clearly, for any wordu we haveA(u) ⊇ F(u) since the difference d = 1 in the definition of an arithmetical subword gives all factors ofu.

Thesubword complexityof a factorial languageF, or of a wordu, is the function equal to the number of distinct elements of F (factors of u) of length n and denoted byf_F(n) (f_u(n)). Thearithmetical complexityofuis defined asa_u(n) = f_A(u)(n). Clearly, a_u(n) ≥ f_u(n) for any word u and length n. We see that both complexities count factors of some factorial language, so, it is not surprising that some techniques useful for computing subword complexity can be used for computing the arithmetical one. In this paper, we shall use one of them which involves bispecial words. For proofs and details of this method, see Cassaigne [6].

Let F be a factorial language. The set of all elements of F of length n will be denoted by F(n). For a word v ∈ F, let us define the set of symbolsR(v) as R(v) = {a ∈ Σ|va ∈ F}. Its cardinality r(v) = #R(v) is called the (right) speciality degreeofv.

Analogously, let us defineL(v) ={a∈Σ|av∈ F}and l(v) = #L(v). A word v is bispecial in F if l(v)= 1 and r(v)= 1. Its bilateral orderb(v) is defined as b(v) = #{(a, b)|a, b∈Σ, avb∈ F} −l(v)−r(v) + 1. It is not difficult to check that b(v) can be not equal to zero only ifv is bispecial in F; so, if b(v)= 0, we say thatv is anessential bispecial word inF. The set of all essential bispecial words of lengthnin F is denoted byB_F(n).

It is convenient to depict words of the formavb ∈ F, where v ∈ F, a, b ∈Σ, as edges of the bipartitebispeciality graph of the wordv whose parts areL(v) and R(v) (see Fig. 1). The bilateral order ofv is thus the number of edges of this graph, minus number of its vertices, plus 1.

Let us denote byf(n) (f(n)) the first (second) difference of a functionf(n):

f(n) =f(n+ 1)−f(n),f(n) =f(n+ 2)−2f(n+ 1)−f(n). We know [6] the

following relations for complexity of a factorial languageF:

f_F (n) =

v∈F(n)

(r(v)−1), (1)

f_F(n) =

v∈F(n)

b(v).

By the definition of the set of essential bispecial words, in the latter formula we can ignore all words but them:

f_F(n) =

v∈BF(n)

b(v). (2)

At last, we mention another pair of easy relations valid for any infinite worduand anyn >0 (recall thatq= #Σ):

f_u(n)

q ≤f_u(n−1)≤f_u(n). (3)

2. Patterns and Toeplitz words

Let ?∈/Σ be a new symbol calledgap. Apatternis a finite word on Σ∪{?}. For a patternTand an infinite wordu=u1u2· · ·u_n· · ·,u_i∈Σ∪{?}, we shall denote by T·uthe result of substituting symbols ofuto successive gaps ofT^ω(=T T· · ·T· · ·).

If all patternsT1,T2, . . . , T_n, . . .start with a symbol of Σ, then there exists a limit t=T1·T2·· · ··Tn·· · · ∈Σ^ωof the sequenceT1·(T2·· · ··(Tn·?^ω)· · ·). The sequence t is called the Toeplitz word generated by the sequence of patterns{Ti}^∞_i=1. For studies of Toeplitz words of specific forms, see [9, 20].

Note that each of the words T1·(T2· · · · ·(T_n·?^ω)· · ·) is periodic, and if we naturally denote the pattern corresponding to its minimal period byT1·T2·· · ··T_n, we obtain an associative operation (·) on patterns.

Example 2.1. Let T = a?b?, then T·?^ω = a?b?a?b?a?b?a?b?· · ·, T·(T·?^ω) = aab?abb?aab?abb?· · ·, and thus T·T =aab?abb?. Continuing the process, we get the famouspaperfolding word t=T ·T ·T · · · ·=aabaabbaaabbabba· · ·. Both its subword [2] and arithmetical [5] complexities are linear.

Let us fix a primepand a letter a∈Σ and define the patternT_p(a) =a · · ·a

p−1

?.

The patternT_p(a1)·T_p(a2)·· · ··T_p(a_n) will be denoted byT_p(a1a2· · ·a_n) for short;

its length ispⁿ and the only gap in it is the last symbol. The word on Σ obtained fromT_p(a1a2· · ·a_n) by erasing the last gap will be denoted by t_p(a1a2· · ·a_n). It is natural to defineT_p(λ) = ? andt_p(λ) =λ.

Example 2.2. SinceT2(a) =a? and T2(b) =b? (and thust2(a) =aandt2(b) = b), we have T2(aa) = aaa? and t2(aa) = aaa, then T2(aab) = aaabaaa? and t2(aab) =aaabaaa, etc.

Now let us fix an infinite word u=u1u2· · ·u_n· · ·, where u_i ∈ Σ, and define the Toeplitz word

t_p(u) =T_p(u1)·T_p(u2)· · · · ·T_p(u_n)· · ·:

each symbol of this infinite word is well-defined since the firstpⁿ−1 symbols are determined already int_p(u1· · ·u_n). Equivalently,t_p(u) can be defined as the limit of the sequence{t_p(u1· · ·u_n)}n; it always exists.

Example 2.3. The famous period doubling word (see, for example, [10]) is de- fined as

t2(ababab· · ·) =abaaabababaaabaaabaaabab· · ·.

The subword complexity of t_p(u) is linear for allu[20]. As for the arithmetical one, the word t_p(u) is what we need to prove Theorem 0.1: in what follows, we show thata_tp(u)(n) = Θ(nf_u(log_p(n))).

3. Arithmetical closure of t

(u)

Consideru=u0u1u2· · ·u_n· · ·, whereu_i ∈ Σ. Without loss of generality, we assume that each symbol of Σ occurs inuat least once. In what follows, we denote by Suthe shift Su = u1u2u3· · ·u_n· · ·; correspondingly, S^ku = u_kuk+1uk+2· · · for allk.

Let us count symbols of t_p(u), unlike those of u, starting with 1 not 0, i.e., t_p(u) = w1w2· · ·w_m· · · for w_i ∈ Σ. We say that a position in a Toeplitz word t_p(u), and the symbol occupying it, are ofnth order if the index of this position int_p(u) is divisible bypⁿ, or, equivalently, if inT_p(u0u1· · ·u_n−1)·?^ω there is still a gap at this position. In particular, all symbols oft_p(u) are of order 0. We shall say that theexact order of a position, or a symbol occupying it, is equal tonif it is of ordern but not of ordern+ 1. By the definition of t_p(u), for all n ≥0 all symbols of exact ordernin it are equal tou_n.

Lemma 3.1. For anyu∈Σ^ω and primepthe arithmetical closure oft_p(u)is

A(tp(u)) = ∞ n=0

F(tp(Sⁿu))

x∈Σ

x^∗

.

Example 3.2. If u = abaaab· · ·, then the arithmetical closure of t2(u) is the union of sets of factors of t2(u) = t2(abaaab· · ·) = (abaa)⁷abab· · ·, t2(Su) = t2(baaab· · ·) = (ba)⁷bb· · ·, t2(S²u) =t2(aaab) = a⁷ba⁷· · ·, etc., plus arbitrarily large powers ofaandb.

Proof. Let us use the notationt_p(u) =w1w2· · ·w_n· · ·, wherew_i∈Σ, and denote byw(d, k) the arithmetical subsequence of t_p(u) of difference dstarting withw_k: w(d, k) =w_kw_k+d· · ·w_k+nd· · ·.

First suppose that (d, p) = 1. Then for eachm, exactly one of the firstp^msym- bols ofw(d, k) (say,w_k+nmd for somen_m∈ {1, . . . , p^m}) is ofmth order int_p(u).

It is not difficult to see that the factor of w(d, k) of length p^m−1 starting from its next symbol,i.e., wk+(n_m+1)dwk+(n_m+2)d· · ·wk+(n_m+p^m−1)d, coincides with a prefix oft_p(u). Since mis arbitrary, t_p(u) andw(d, k) thus have arbitrarily long common factors. Butt_p(u) is by constructionuniformly recurrent[9], that is, each of its factors occurs an infinite number of times with bounded gaps. By a (folk- lore) lemma from [5], its arithmetical subsequencew(d, k) is thus also uniformly recurrent.

It is well-known also that if languages of factors of two uniformly recurrent words have an infinite intersection, then they coincide. Indeed, letR_x(n) denote the recurrence function of an infinite wordx, that is the maximal distance between successive occurrences of the same word of lengthninx. Ifxandy are uniformly recurrent and contain a common factor of length max(R_x(n), R_y(n)) +n, then the sets of their factors of length ncoincide. So, if they contain arbitrarily long common factors, then the sets of its factors coincide.

We can conclude thatF(w(d, k)) =F(tp(u)).

Now suppose that (d, p)= 1; sincepis prime, this implies thatp^m|dfor some m > 0. Suppose that m is maximal (i.e., p^m+1 |d) and define m ≥ 0 as the maximal exponent ofpdividingk: p^m|k, p^m+1 |k. Ifm ≥m, then all symbols of w(d, k) are of order (at least) m in t_p(u), and w(d, k) is a subsequence of w(p^m, p^m) = t_p(S^mu) of difference d/p^m coprime with p. Analogously to the previous paragraph, F(w(d, k)) = F(tp(S^mu)). If m < m, then all symbols of w(d, k) are of exact orderm in t_p(u), and clearlyw(d, k) =u^ω_m. For differentm we obtain all symbols of Σ which occur inu.

We listed all possible cases for dand k, and the observation that A(tp(u)) =

∪k,d>0Fw(d, k) completes the proof of the lemma.

In what follows, we are going to characterize essential bispecial words of the languageA=A(tp(u)). It will help us to estimate its subword complexity, which is the arithmetical complexity oft_p(u): f_A(n) = a_tp(u)(n). In what follows, the notationsf_A(n) anda_tp(u)(n) are used equivalently.

4. Bispecial words of A

Let us define the Toeplitz product of a pattern of the form T = t?, t ∈ Σ^∗, and a wordv=v1· · ·v_n,v_i ∈Σ, as the wordT·v=tv1tv2t· · ·tv_nt. Clearly, the length ofT ·v is|T·v|= (|t|+ 1)(|v|+ 1)−1.

Example 4.1. For each word v =v1· · ·v_n with v_i ∈ Σ we have ?·v =v and T_p(v1· · ·v_n−1)·v_n =t_p(v). For each patternT =t? and for allx, y ∈Σ we have T·λ=t,T ·x=txt,T·xy=txtyt.

Now consider a wordv∈ F(u) and denote bya_vits last symbol. We shall write v=va^σ_v^(v), wherev is the longest prefix of vwhich does not end witha_v. Thus, σ(v) is defined as the length of the longest suffix of v in which only one letter occurs. We shall call the wordv saturatedin F(u) if v ∈ F(u) butva_v ∈ F/ (u), i.e., ifv cannot be extended to an element ofF(u) by repeating its last symbol.

For technical reasons, we for a while assert that

all symbols of Σoccur inuinfinitely many times. (4) Later, in the end of Section 6, we will turn this condition down, but at the moment it is convenient to use it.

Lemma 4.2. Let p ≥ 3 be a prime integer, and u ∈ Σ^ω be an infinite word satisfying Condition (4). Then each essential bispecial word of the arithmetical closureAof the Toeplitz wordt_p(u)is of one of the following four forms:

(1) t_p(v), where v∈ F(u);

(2) T_p(v)·a^2p_v ^σ(v)−2, wherev=va^σ_v^(v)is a saturated word inF(u)withv=λ;

(3) T_p(v)·x=t_p(v)xt_p(v), wherex∈Σ,v∈ F(u);

(4) T_p(v)·xy=t_p(v)xt_p(v)yt_p(v), where x, y∈Σ,x=y,v∈ F(u).

Proof. Let us consider a bispecial wordw∈ A. Due to Lemma 3.1, it must belong to someF(t_p(Sⁿu)): indeed, even ifwis a power of a symbolx, we must be able to extend it not only byx, so it must occur somewhere else than inx^∗.

First of all, let us prove that

w=T_p(v1· · ·v_k)·z (5)

for some k≥ 0 and z which is either of length not greater than 2 or is a power of a letterx; and each wordSⁿusuch thatw∈ F(tp(Sⁿu)) starts withv1· · ·v_k. Indeed, if|w| ≤2 orw=x^mfor some letterxand integerm, then it is true with k= 0 (since T_p(λ)·w=?·w=w). So, suppose that |w| ≥3. Sincew is a factor of somet_p(Sⁿu), where Sⁿu starts with a letter v1, and sincew contains other symbols thanv1, it looks like

w=v^α₁x1v₁^p−1x2· · ·x_mv^β₁,

where 0≤α, β≤p−1,x1· · ·x_m∈ F(tp(Sⁿ⁺¹u)), and at least one of the symbols x_i is not equal tov1. We shall denote this symbol byx. Since|w| ≥3,wcontains at least one of the factorsv1v1x,v1xv1, orxv1v1. Hencewcannot be a factor of any t_p(Sⁿu) whereSⁿustarts with a symboly=v1because in suchSⁿuoccurrences of any symbol butyare separated byp−1 symbolsy. (Note that here we use the conditionp >2: ifp= 2, arbitrarily long words of the formv1yv1yv1y· · · occur both int2(v1yy· · ·) and in t2(yv1v1· · ·), complicating the situation.)

Thus,w occurs in Aonly as a factor of some words of the form t_p(v1· · ·). A symbol not equal to v1 may occur in it, or in its extension, only at a distance divisible by p from x. In particular, if α = p−1 (β = p−1), then w can be extended to the left (to the right) inAonly byv1. Butwis by assertion bispecial, so,α=β =p−1 and thus w=T_p(v1)·(x1· · ·x_m). Herex1· · ·x_m is a factor of t_p(Sⁿ⁺¹u· · ·) for allnsuch that wis a factor oft_p(Sⁿu). If the lengthm≤2 or x1=x2=· · ·=x_m, then (5) is proved withk= 1, otherwise we can continue the process withx1· · ·x_minstead ofw: it is also bispecial inAsince it can be extended

at least by the same letters thanw. After several steps we shall necessarily come to (5).

If|z|= 2 and z =xy, x=y, we get exactly Case 4. In all other cases z is a power of some symbol: z=a^m, a∈Σ,m≥0.

If k = 0 andm = 0, we have w =λ =t_p(λ), so that w falls into Case 1. If k >0, thenm >0: we cannot havez=λsinceT_p(v)·λ=t_p(v) =T_p(v)·a^p_v^σ(v)−1, and if w = t_p(v) we would stop the process earlier, at the representation w = T_p(v)·a^p_v^σ(v)−1(with a=a_v,m=p^σ(v)−1). So, from now on we consider that m >0.

Also,v_k =a: otherwise we also would stop the process earlier, at most at the representationw=T_p(v1· · ·v_k−1)·a^p(m+1)−1.

Let us mention that the greatest power ofaoccurring inF(t_p(aⁱb· · ·)), where b = a, is a^2pi−1: indeed, a^2pi−1 =T_p(aⁱ)·a; the letter a will somewhere occur between twot_p(aⁱ) because of Condition (4).

So, suppose first that m = pⁱ −1 (or m = 2pⁱ − 1) with i > 0; then, due to the previous paragraph, w occurs in A if and only if v = v1· · ·v_kaⁱ ∈ F(u); here v1· · ·v_k = v, a = a_v and i = σ(v). In the first situation, w = T_p(v1· · ·v_k)·a^m=t_p(v1· · ·v_kaⁱ), and this gives us exactly Case 1; in the second one,w =T_p(v1· · ·v_k)·a^m = T_p(v1· · ·v_kaⁱ)·a=t_p(v1· · ·v_kaⁱ) a t_p(v1· · ·v_kaⁱ).

This situation falls into Case 3; to describe this case completely, we add the pos- sibility ofi= 0, that is, ofm= 1 and z=a.

It remains to consider the casez=a^m, wherem=pⁱ−1,m= 2pⁱ−1 for any i, andm≥2. As we have mentioned,w=T_p(v1· · ·v_k)·a^m occurs inAonly as a factor of somet_p(vaⁱ· · ·), where 2pⁱ⁻¹−1< m≤2pⁱ−1.

Suppose that bw∈ Afor some b=a. Then bw occurs as a factor in words of the form t_p(v1· · ·v_kaⁱ· · ·) and only in them, moreover, the first b of bw always occurs in such a word at a position of orderk+i, i.e., is always followed by the word t_p(v1· · ·v_kaⁱ) of length p^k+i−1, then by a symbol (of order k+i again), then byt_p(v1· · ·v_kaⁱ) again etc. But we assumed thatm=pⁱ−1 andm= 2pⁱ−1 for anyi, and thus |bw|= 1 +|Tp(v1· · ·v_k)·a^m|=p^k(m+ 1) is not divisible by p^k+i. So, the wordbwcan be extended to the right by only one of the letters; its exact order int_p(v1· · ·v_kaⁱ· · ·) lies betweenk andk+i−1, so it is equal toa.

The symbolb=acan be chosen arbitrarily by Condition (4), so, for all b=awe havebwa∈ A,bwc /∈ Afor anyc=a.

Analogously,awb∈ Afor allb=a(andcwb /∈ Afor allb, c=a, but we already know this because of the symmetry ofb andc). So, to find the bilateral order of w it remains to find out if awa∈ A: this word must be considered individually since we cannot determine the position of its first (or last) symbol modulo p^k+i, we know only that it is of order k in t_p(v1· · ·v_kaⁱ· · ·) and thus that awa may occur inAonly as a factor ofT_p(v1· · ·v_k)·a^m+2=t_p(v1· · ·v_k)awat_p(v1· · ·v_k).

Ifawa∈ A, then the bispeciality graph ofwlooks as it is depicted in Figure 2;

it has 2q−1 edges and 2qvertices, so, b(w) = 0 andw is not essential bispecial.

Ifawa /∈ A, then the bispeciality graph of wlooks as it is depicted in Figure 3, andb(w) =−1, so,wisessential bispecial. But this is the case only if, first, iis

Figure 2. b(w) = 0. Figure 3. b(w) = −1 (Case 2).

the maximal power ofawhich can followv1· · ·v_k in u(i.e., if the wordv1· · ·v_kaⁱ is saturated), and second, if m = 2pⁱ−2, that is, if the maximal number of consequtivea’s in allt_p(Sⁿu) of the formt_p(S^kv1· · ·v_kaⁱ· · ·) is exactlym+ 1, so thatT_p(v1· · ·v_k)·a^m+2 ∈ A. Moreover, the case/ k= 0 (i.e., ofv1· · ·v_k =λand w=a^2pi−2) is excluded since any power ofaincludinga^2pi always belongs to A due to Lemma 3.1. We have obtained exactly Case 2 of Lemma 4.2 and completed

its proof.

5. Bilateral orders and diagram of first differences

As announced at the end of the introduction, we continue the proof of Theo- rem 0.1 finding bilateral orders of all essential bispecial words listed in Lemma 4.2.

We shall consider all the four cases and will find second differences of the subword complexity function. Let us start from

Case 1. Consider a word of the formw=t_p(v), wherev∈ F(u),|v|=m. Note thatw occurs inA only as a factor of words of the formt_p(v· · ·). Suppose that a_v = a, that is, w= T_p(v)·a^pk−1 for some k >0, where v does not end with a. By the construction, all symbols of thisa^pk−1 are always of order |v| in any infinite wordt_p(v· · ·). Let us consider possible extensionsbwcofwinA.

If we choose an arbitraryb=a, we see that the firstb ofbwcmust be of order

|v| =m (in any occurrence of bw in t_p(v· · ·)). If vb ∈ F(u), then the order of b can be equal exactly to m, and thus the order of the last cof bwc∈ A can be arbitrarily large. Due to (4),c can be arbitrary, so,bwc∈ Afor allb∈R(v)\{a}, c ∈Σ (and, symmetrically, for all b ∈ Σ, c ∈ R(v)\{a}). On the other hand, if b /∈R(v), then the order of the first b of bwcmust be greater than m, the order of the lastcmust be equal exactly tom, and thusvc must belong toF(u). So, if b=a(or, symmetrically,c=a), andb, c /∈R(v), thenbwc /∈ A.

It remains to find out if awa ∈ A. If a ∈ R(v), this is the case by the same arguments as above: the exact order of the first and the lastacan be equal tom.

But this is the case even ifa /∈R(v),i.e., ifvis a saturated word inF(u). Indeed, in this case all p^k+ 1 symbols of order |v| in awa are equal toa. Any of them except the first and the last one may have order |v| (and arbitrarily large exact

a₁

2

1 2

a

q q

r(v)+1

r(v) r(v)

r(v)+1

Figure 4. Case 1,vis not saturated.

1

2

1

2

q

a q a

r(v) r(v)+1

r(v) r(v)+1

Figure 5. Case 1, vis saturated.

order, so, it can be equal toa due to (4)). All other symbols are of exact order less than|v|. So,awa∈ Aanyway.

We see that the form of the bispeciality graph oft_p(v) depends whether v is saturated in F(u) or not (see Figs. 4, 5). The number of edges in the graph is respectivelyqr(v) + (q−r(v))r(v) =r(v)(2q−r(v)) andr(v)(2q−r(v)) + 1, the number of vertices is always 2q, and thus

b(w) = (r(v)−1)(2q−r(v)−1) + 1, ifv is saturated, (r(v)−1)(2q−r(v)−1) otherwise.

As it follows from Lemma 4.2, all essential bispecial words ofAof lengthp^m−1 correspond to words ofF(u) of lengthm. If we denote the set of factors ofuof lengthm byFu(m) and the number of saturated words in Fu(m) bys_u(m), we have, due to (2),

f_A(p^m−1) =s_u(m) +

v∈Fu(m)

(r(v)−1)(2q−r(v)−1).

Case 2. As we have shown in the proof of Lemma 4.2, each saturated wordv of lengthmwithv=λin ugives an essential bispecial wordT_p(v)·a^2p_v ^σ(v)−2 inA.

Its length is (2p^σ(v)−1)p^|v|−1 = 2p^m−p^|v|−1 and its bilateral order is −1.

So, lengths of all such words lie between 2p^m−p^m−1−1 and 2p^m−2 (in fact, the latter value can be specified to 2p^m−p−1 sincev=λ, but we neglect it). Their combined number iss_u(m) minus the number of saturated words withv =λ, that is, of symbols whose maximal power inuism. If we denote the latter number by q_u(m), we get

f_A(2p^m−1)−f_A (2p^m−p^m−1−1) =

2p^m−2 n=2p^m−p^m−1−1

f_A(n) =−s_u(m) +q_u(m).

1

2

1

2

r(v) r(v)

Figure 6. Case 3, x∈L(v).

1

2

1

2

q q

r(v) r(v)+1

r(v) r(v)+1

x x

Figure 7. Case 3, x∈L(v).

Case 3. Consider w = t_p(v)xt_p(v), where v ∈ F(u), |v| = m (and thus |w| = 2p^m−1), x∈Σ. We have to consider two cases, a_v=xanda_v=x.

Suppose first thatvdoes not end withx: a_v=x. Then the order of the central symbolxofwin any of its occurrences in t_p(v· · ·) is (at least)m.

If we assume in addition that vx /∈ F(u), then the exact order of the central xofw must always be greater thanm. So, the symbols situated at the distance p^m from it must be of exact orderm, which means that they both must be equal and must belong toL(v) inF(u). But these two symbols are exactly symbols by whichwcan be extended to the left and to the right in A: the wordwoccurs in eacht_p(va· · ·), where a∈L(v), and can be extended in it only asawa. So, the bispeciality graph ofw looks as it is depicted in Figure 6; it hasr(v) edges and 2r(v) vertices, and the bilateral order ofwisb(w) =−r(v) + 1.

All theser(v) edges by the same reasons occur also whenvx∈ F(u), but in this case, the centralxofwmay have exact ordermin somet_p(vx· · ·). So, ifbwc∈ A for some b, c∈Σ, then the exact order of b can be arbitrarily large, and due to (4),bcan be arbitrary; but ifb=x, this implies that the exact order ofcismand thusc=x. Similarly,ccan be arbitrary with b=x. So, the bispeciality graph of wlooks as it is shown in Figure 7; it has 2q+r(v)−2 edges and 2qvertices, and b(w) =r(v)−1.

Now let us consider x=a_v. In this case, w =T_p(v)·x^2pσ(v)−1, and each of these 2p^σ(v)−1 lettersxis of order (at least)|v|=m−σ(v).

Suppose that vx /∈ F(u); then, as it was mentioned earlier, 2p^σ(v)−1 is the maximal number of successive lettersxof order|v|in a word of the formt_p(v· · ·) whose factors belong to A. So, the central one of them, that is the centralxof w, always is of orderm(and even greater), and we can continue the arguments as in the case ofx=a_v and obtain the bispeciality graph from Figure 6 again. The only difference occurs ifv =x^m and thusw=x^2pm−1: as a factor ofx^ω, it can be extended also as xwx. Its bispeciality graph is depicted in Figure 8, and its bilateral order is−r(v).

Now letvx∈ F(u); then each of the 2p^σ(v)−1 lettersxof order|v|in wmay be of orderm. However, if this is not the central one, i.e., if the centralxinwis of order less thanm, then this occurrence ofwcan be extended only asxwx. On

1

2

1

2

r(v)

r(v)+1 r(v)+1

r(v)

x x

Figure 8. Case 3, v=x^m,x^m+1 ∈ F(v).

1 2

1 2

x x

y y

Figure 9. Case 4.

the other hand, if the centralxis of orderm, all arguments applied forx=a_vare valid. We see that the bispeciality graph ofwlooks again as in Figure 7.

So, for each v ∈ Fu(m) there are r(v) words of length 2p^m−1 in A having bilateral orderr(v)−1 (namely, words T_p(v)·x, x∈ L(v)), andq−r(v) words of the same length of bilateral order −r(v) + 1 or −r(v), where each word of b(w) =−r(v) corresponds to a saturated word of the formv=x^m; the number of such words isq_u(m). Since due to Lemma 4.2 there are no other bispecial words of length 2p^m−1, we have

f_A(2p^m−1) =−q_u(m) +

v∈Fu(m)

(2r(v)−q)(r(v)−1).

Note that this value can be positive, negative, or equal to zero.

Case 4. Consider w = t_p(v)xt_p(v)yt_p(v), where |v| = m; then |w| = 3p^m−1.

Sincex=y,xandyare always of ordermint_p(v· · ·), and the exact order of one of them is even larger. The other symbol must be of exact orderm. Let this symbol bex(so, vx∈ F(u)); thenwcan be extended only asxwxsincep−1 symbols of ordermsurroundingy of order greater thanmare ofexactorderm(here we once again use the conditionp≥3). Ifvy /∈ F(u), thenw is not bispecial at all, but otherwise symmetricallyywy∈ A. So, for each pair of distinct lettersx, y ∈L(v) the respective wordwhas the bispeciality graph depicted in Figure 9; its bilateral order is−1. The number of such pairsx, y for a givenvisr(v)(r(v)−1), so,

f_A(3p^m−1) =−

v∈Fu(m)

r(v)(r(v)−1).

We have found all values off_A(n) forn∈ {p^m−1, . . . , p^m+1−1}. Figure 10 shows the behaviour off_A(n) at this interval.

The rectangle shows the interval from 2p^m−p^m−1−1 to 2p^m where f_A (n) somehow like stairs decreases by s_u(m). (Recall that s_u(m) is the number of saturated words of length m in u; the notation is introduced in the end of the consideration of Case 1.) All other non-zero values off_A(n) are explicitly found above.

Figure 10. The behaviour of f_A(n).

6. Estimates for arithmetical complexity

In this section, we end the proof of Theorem 0.1 estimating the arithmetical complexity. First, still under Condition (4), we shall sum up the second differences.

According to the results of the previous section,

f_A(p^m+1−1)−f_A (p^m−1) =

p^m+1−2

n=p^m−1

f_A(n)

=f_A(p^m−1)−s_u(m) +q_u(m) +f_A(2p^m−1) +f_A(3p^m−1)

= (q−1)

v∈Fu(m)

(r(v)−1) = (q−1)f_u(m).

This is valid starting from m = 0: indeed, there is one word λ in Fu(0), it is not saturated since it can be extended to the right by any symbol, and can be considered as well as any other word ofF(u). Clearly,f_A (0) =q−1, so,

f_A (p^m+1−1) = m n=0

(q−1)f_u(n) +f_A (0) = (q−1)(f_u(m+ 1)−f_u(0) + 1)

= (q−1)f_u(m+ 1).

This value is valid also for alln∈ {3p^m, . . . , p^m+1−1}; forn∈ {2p^m, . . . ,3p^m−1}

we have f_A (n) := B2 = (q−1)f_u(m+ 1) +

v∈Fu(m)r(v)(r(v)−1), and for n ∈ {p^m, . . . ,2p^m−1}, the value of f_A (n) lies between B1 := (q−1)f_u(m) +

v∈Fu(m)(r(v)−1)(2q−r(v)−1) and B1+f_u(m) (since 0≤s_u(m) ≤f_u(m)).

So,

f_A(p^m+1)−fA(p^m) =

p^m+1−1

n=p^m

f_A (n)≥p^mB1+p^mB2+(p^m+1−3p^m)(q−1)fu(m+1)

=p^m[(pq−p+ 1)f_u(m+ 1)−qf_u(m)]≥p^m(p−1)(q−1)f_u(m+ 1).

Similarly,

f_A(p^m+1)−f_A(p^m)≤p^m(B1+f_u(m)) +p^mB2+ (p^m+1−3p^m)(q−1)f_u(m+ 1)

≤p^m

pq−p+ 1−q−1 q

f_u(m+ 1).

Summing up, we see that f_A(p^m+1) =

m n=0

[f_A(pⁿ⁺¹)−f_A(pⁿ)] +q,

and using the inequalities above and again (3), we obtain that (p−1)(q−1)q

pq−1 fu(m+ 1)(p^m+1−1)< fA(p^m+1)≤pq²−pq+ 1

q(p−1) fu(m+ 1)(p^m+1−1) +q.

Note that for alln∈ {p^m, . . . , p^m+1−1}the inequalityf_A (n)≥(q−1)f_u(m+ 1) holds. So, sincep^m≤n, we see that

f_A(n+ 1)≥f_A(p^m) + (n−p^m)(q−1)f_u(m+ 1)> f_u(m+ 1)

(q−1)(p−1) pq−1 n−1

and

f_A(n+ 1) ≤ f_A(p^m+1)−(p^m+1−n−1)(q−1)f_u(m+ 1)

≤ f_u(m+ 1)

q(q−1)(2p−1) +p

q(p−1) n−q²−q+ 1 q(p−1)

+q.

Sincem+ 1 =log_p(n+ 1), the latter two inequalities give us the statement of Theorem 0.1 under Condition (4):

(C1n−1)f_u(log_p(n+ 1))< f_A(n+ 1)≤(C2n−C3)f_u(log_p(n+ 1)) +q, (6) whereC1=^(q−_pq−^1)(p−₁ ¹⁾,C2=^q(q−1)(2_q(p−^p−₁₎^1)+p,C3= ^q_q(p−^2−q+1₁₎· It remains to turn down Condition (4). Let us consider an infinite wordunot satisfying it and denote by Σthe set of letters which occur in it an infinite number of times. Let the last occurrence of a letter from Σ\Σ be the Mth letter of u;

then u = S^Mu satisfies (4) and thus Theorem 0.1 is valid for it: a_tp(u)(n) = Θ(nf_u(log_pn)). But for all n ≥ M, we clearly have f_u(n) = f_u(n) +M, so, Θ(nf_u(log_pn)) = Θ(n(f_u(log_pn) +M)) = Θ(nf_u(log_pn)) since f_u

is a positive non-decreasing function. Then, due to Lemma 3.1, A(tp(u)) =

∪^M−_n=0¹F(tp(Sⁿu)) ∪ A(tp(u)) ∪_x∈Σ\Σ x^∗, and thus a_tp(u)(n) ≤ a_tp(u)(n) ≤ a_tp(u)(n) +_M−1

n=0 f_Sⁿ_u(n) + #(Σ\Σ). Since each off_Sⁿ_u(n) grows linearly [20], we havea_tp(u)(n)≤a_tp(u)(n) +Dn+q for someD andq. Sincea_tp(u)grows at least as Θ(n), this impliesa_tp(u)= Θ(a_tp(u)). So, since the theorem holds foru, it holds also foru. The proof of Theorem 0.1 is complete.