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  • ACTA ARITHMETICA

    LX.1 (1991)

    Generalized Rudin–Shapiro sequences

    by

    Jean-Paul Allouche (Talence) and Pierre Liardet* (Marseille)

    1. Introduction

    1.1. The Rudin–Shapiro sequence was introduced independently by these two authors ([21] and [24]) and can be defined by

    εn = (−1)u(n) ,

    where u(n) counts the number of 11’s in the binary expansion of the integer n (see [5]). This sequence has the following property:

    (1) ∀N ≥ 0, sup θ∈R

    ∣∣∣ ∑ n

  • 2 J.-P. Allouche and P. Liardet

    exponent α(x) is explicitly given and satisfies

    ∀x 1/2 ≤ α(x) ≤ 1 , ∀x 6∈ Z α(x) < 1 , ∀x ∈ Z + 1/2 α(x) = 1/2 .

    Moreover, the constant C ′ does not depend on x. Of related interest see the papers of Rider [20], Brillhart and Carlitz [5],

    Brillhart and Morton [7], Brillhart, Erdős and Morton [6], Mendès France and Tenenbaum [17], Queffélec [18], and Boyd, Cook and Morton [4].

    1.2. In this paper we are going to extend these results to other se- quences. One particularly appealing example of generalization consists in counting the number of words of length d + 2 which begin and end in 1 in the binary expansion of the integer n (the case d = 0 gives precisely the Rudin–Shapiro sequence). Extending this idea we will introduce Hadamard matrices: such a matrix (of order q) gives sequences which can be generated by finite automata and which satisfy (2) where M2 is replaced by the set Mq of q-multiplicative sequences of modulus 1 (for the theory of Hadamard matrices, see for example [25]).

    Section 2 is devoted to a general notion of sequences (called chained sequences) in a compact and metrizable group. These sequences satisfy inequalities analogous to (2) where the exponential function is replaced by an irreducible representation of the group. When the group is abelian, the optimal case giving bounds as in (1) implies that the group is necessarily finite. Let us notice (from Lemma 4 below) that bounds as in (1) and (2) depend only on the orbit of the sequence under the shift: for instance every sequence in the closed orbit of the Rudin–Shapiro sequence under the shift on {−1,+1}N satisfies (1) where C is replaced by another suitable constant.

    In Section 3 we define generalized Rudin–Shapiro sequences including previous extensions introduced by M. Queffélec ([19]). These sequences still have the Lebesgue measure as spectral measure.

    2. Chained sequences

    2.1. Notations and definitions. In what follows q is an integer greater than or equal to 2, A is the alphabet {0, 1, 2, . . . , q − 1} with the natural order. The monoid of finite words on A is denoted by A∗ and is ordered with the lexicographical order, denoted by ≤. The number of letters in a word w is called the length of w and denoted by |w|. The empty word Λ is of length 0. Let Ar be defined by

    Ar := {w ∈ A∗; |w| = r} . Let d be a positive integer and D := Ad+1. The set D can be considered as

  • Rudin–Shapiro sequences 3

    an alphabet ordered by the lexicographical order. There exists a canonical one-to-one order-preserving map c from D∗ into A∗ which identifies D and Ad+1.

    Expanding an integer in base q allows us to define the maps ek (“kth digit”) from N to A by

    n = ∑ k∈N

    ek(n)qk .

    We denote by ñ the word et(n)(n) . . . e0(n), with t(0) = 0, and for n 6= 0, t(n) = [(log n)/(log q)] ([x] is the integral part of x). Moreover, to each word w = wr−1 . . . w0 in A∗ (r ≥ 1) we associate the integer

    ẇ = qr−1wr−1 + . . . + qw1 + w0 ,

    and we define Λ̇ = 0. For every sequence ϕ with values in a set E, we define ϕ̇ : A∗ → E by

    ϕ̇(w) = ϕ(ẇ) ,

    and for every map f from A∗ to E we define the sequence f̃ with values in E by

    f̃(n) = f(ñ) .

    Definition 2.1. Let G be a multiplicative group. A map f from A∗ to G is called a chained map (over A) if for all letters a, b ∈ A and every word w in A∗

    (i) f(0w) = f(w), (ii) f(abw) = f(ab)f(b)−1f(bw).

    (The rôle of property (i) is to select chained maps f which derive from a sequence ϕ by the relation ϕ̇ = f . If this property is explicitly required we shall speak of regular chained map. Otherwise we shall omit this extra condition.)

    If G is equal to C\{0} (resp. R), f will be called a multiplicative chained map (resp. an additive chained map). Note that iterating (ii) gives for all letters a1, . . . , as in A and every word w in A∗

    (3) f(a1 . . . asw) = (f(a1a2)f(a2)−1) . . . (f(as−1as)f(as)−1)f(asw) .

    An easy computation yields

    (4) f(αβγ) = f(αβ)f(β)−1f(βγ)

    for all nonempty words α, β, γ in A∗.

  • 4 J.-P. Allouche and P. Liardet

    Finally, if G is abelian one has for every positive integer n

    (5) f̃(n) = f(0) {∏

    k∈N (f(ek+1(n)))−1f(ek+1(n)ek(n))

    } .

    By (4) the map f which is chained over A can be lifted through the canonical map c (from D∗ = (Ad+1)∗ to A∗) to a map defined on D∗ and chained over D.

    A sequence ϕ with values in G is called chained to base q (or chained over A) if the associated map ϕ̇ (from A∗ to G) is chained. Note that ϕ̇ is then a regular chained map to base qd+1 for every positive integer d.

    Example 1. Let ϕ be a complex sequence such that ϕ(0) = 1 and satisfying the functional equation

    ∀ k ∈ N, ϕ(n) = ∏ k∈N

    ϕ(ek(n)) .

    Then ϕ is said to be strongly q-multiplicative (for q-multiplicativity, see [11]). This sequence is multiplicatively chained to base q. Actually, the map ϕ̇ is a morphism from the monoid A∗ to the multiplicative group C \ {0}.

    Example 2. Let v be a word in A∗ of length r + 1 (r ≥ 0). We denote by 0r the word consisting of r letters 0 (if r = 0 we define 00 = Λ). Let Zv(w) be the number of occurrences of the word v in the word 0rw. This number depends only on the integer n = ẇ if v 6= 0r+1, and we will also denote it by Zv(n).

    Proposition 2.1. Let v be a word in A∗ such that |v| = r + 1 and v 6= 0r+1. Then the sequence (Zv(n))n is additively chained to base qs for every integer s ≥ max{1, r}.

    P r o o f. If r = 0 one has, for all words w and w′ in A∗, Zv(ww′) = Zv(w)+Zv(w′), hence the result. Suppose r ≥ 1. Let S = As for s > r and let c be the canonical morphism from S∗ to A∗. We have to prove that the map Zv ◦ c from S∗ to R is additively chained over S.

    Let α and β be letters in S and w be a word in S∗. The integer Zv(c(α)c(β)c(w)) is equal to the number of occurrences of v in the words 0sc(α)c(β) and 0sc(β)c(w) minus the number of occurrences of v which:

    — either occur at the same places in the words c(α)c(β) and 0sc(β), — or occur in 0sc(β) but do not occur in c(α)c(β).

    This last number is precisely Zv(c(β)), hence

    Zv ◦ c(αβw) = Zv ◦ c(αβ)− Zv ◦ c(β) + Zv ◦ c(βw) . In particular, the sequence u discussed in the introduction (which corre-

    sponds to Z11(·) with q = 2), is additively chained to base 2.

  • Rudin–Shapiro sequences 5

    Example 3. The following proposition generalizes the previous case:

    Proposition 2.2. Let d be a positive integer and let ϕ be a periodic sequence with values in G such that the period of ϕ is qd+2, and ϕ(0) = 1G (the unity of the group G). Let x ∈ G. Then the sequence

    (6) f(n) = xϕ([n/qt(n)]) . . . ϕ([n/q])ϕ(n)

    is chained to base qd+1.

    P r o o f. For every word w of length r in A∗, say w = wr−1 . . . w0, one has by definition (6)

    ḟ(w) = xϕ̇(wr−1)ϕ̇(wr−1wr−2) . . . ϕ̇(wr−1 . . . w0) .

    Note that the value of ϕ̇(w) depends only on the word wd+1 . . . w0 (w being replaced by 0d+2w if r ≤ d + 1). To prove that ḟ ◦ c is chained (where c is the canonical map from D∗ = (Ad+1)∗ to A∗) it suffices to check that (ii) (in Definition 1) holds for a and b in Ad+1, which is a straightforward computation.

    This proposition leads to a particular case of chained sequence:

    Definition 2.2. Let d ∈ N. A sequence f with values in G is called a d-sequence in base q if there exists a sequence ϕ : N → G such that for every integer n

    (a) ϕ(qn) = 1G, (b) ϕ(n + qd+2) = ϕ(n) (i.e. ϕ is qd+2-periodic),

    and

    (7) f(n) = f(0)ϕ([n/qt(n)]) . . . ϕ([n/q])ϕ(n) .

    By Proposition 2.2 a d-sequence in base q is chained to base qd+1. Choose x in G and v in Ad+2, with v 6= 0d+2. The sequence

    (8) fv(n) = xZv(n)

    is chained to base qd+1 from Proposition 2.1 It is a d-sequence in base q if and only if the last letter of the word v is not 0 (i.e. if v 6∈ A∗0). The map ϕ corresponding to fv by (7) is the characteristic function χv of the arithmetic progression ṽ + qd+2N.

    2.2. Chained sequences in an abelian group. Suppose that the group G is abelian. The set of chained sequences over A with values in G is a commutative group for the usual multiplication:

    fg(n) = f(n)g(n) for every integer n .

    The subset of d-sequences in base q is a subgroup, generated by the d- sequences (8). More precisely, if f is a d-sequence in base q, one easily

  • 6 J.-P. Allouche and P. Liardet

    obtains

    f(n) = f(0) ∏

    v∈Ad+2 (ϕ(v))Zv(n), where ϕ(v) = f(v̇)f(0)−1 .

    In the general case we have

    Theorem 2.1. Let G be an abelian group and let F : A∗ → G be a regular chained map over A. Then

    (9) F (·) = F (0) ∏

    b∈A2,b 6=00

    (F (b))Zb(·) .

    P r o o f. Let h : A∗ → G be defined by the right han