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43
Sums of digits, overlaps, and palindromes
 Discrete Math. & Theoret. Comput. Sci
"... Let ¦¨§�©��� � denote the sum of the digits in the base � representation of �. In a celebrated paper, Thue showed that the infinite word ©� ¦ ¥ ©������������������� � is overlapfree, i.e., contains no subword of the form �������� � , where � is any finite word and � is a single symbol. Let ���� � ..."
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Let ¦¨§�©��� � denote the sum of the digits in the base � representation of �. In a celebrated paper, Thue showed that the infinite word ©� ¦ ¥ ©������������������� � is overlapfree, i.e., contains no subword of the form �������� � , where � is any finite word and � is a single symbol. Let ���� � be integers with ���� � , ���� �. In this paper, generalizing Thue’s result, we prove that the infinite word �¨§� � �� � ��©�¦¨§�©������������� � ���� � is overlapfree if and only if ���� �. We also prove that ��§¨ � � contains arbitrarily long squares (i.e., subwords of the form �� � where � is nonempty), and contains arbitrarily long palindromes if and only if ���� �.
Doppler resilient Golay complementary pairs for radar,” presented at the
 IEEE Statist. Signal Process. Workshop (SSP
, 2007
"... We present a systematic way of constructing a Doppler resilient sequence of Golay complementary waveforms for radar, for which the composite ambiguity function maintains ideal shape at small Doppler shifts. The idea is to determine a sequence of Golay pairs that annihilates the loworder terms of th ..."
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We present a systematic way of constructing a Doppler resilient sequence of Golay complementary waveforms for radar, for which the composite ambiguity function maintains ideal shape at small Doppler shifts. The idea is to determine a sequence of Golay pairs that annihilates the loworder terms of the Taylor expansion of the composite ambiguity function. The ProuhetThueMorse sequence plays a key role in the construction of Doppler resilient sequences of Golay pairs. We extend this construction to multiple dimensions. In particular, we consider radar polarimetry, where the dimensions are realized by two orthogonal polarizations. We determine a sequence of twobytwo Alamouti matrices, where the entries involve Golay pairs and for which the matrixvalued composite ambiguity function vanishes at small Doppler shifts. 1.
A Unique Decomposition Theorem for Factorial Languages
 Internat. J. Algebra Comput
"... We study decompositions of a factorial language to catenations of factorial languages and introduce the notion of a canonical decomposition. ..."
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We study decompositions of a factorial language to catenations of factorial languages and introduce the notion of a canonical decomposition.
Arithmetical Complexity of Infinite Words
"... We introduce a new notion of the arithmetical complexity of a word, that is the number of words of a given length which occur in it in arithmetical progressions. The arithmetical complexity is related to a wellknown function of subword complexity and cannot be less than it. However, our main result ..."
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We introduce a new notion of the arithmetical complexity of a word, that is the number of words of a given length which occur in it in arithmetical progressions. The arithmetical complexity is related to a wellknown function of subword complexity and cannot be less than it. However, our main results show that the behavour of the arithmetical complexity is not determined only by the subword complexity growth: if the latter grows linearly, the arithmetical complexity can increase both linearly and exponentially. To prove it, we consider a family of D0L words with high arithmetical complexity and a family of Toeplitz words with low one. In particular, we nd the arithmetical complexity of the ThueMorse word and of the paperfolding word. 1
Behavioural Differential Equations and Coinduction for Binary Trees
"... Abstract. We study the set TA of infinite binary trees with nodes labelledinasemiringA from a coalgebraic perspective. We present coinductive definition and proof principles based on the fact that TA carries a final coalgebra structure. By viewing trees as formal power series, we develop a calculus ..."
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Abstract. We study the set TA of infinite binary trees with nodes labelledinasemiringA from a coalgebraic perspective. We present coinductive definition and proof principles based on the fact that TA carries a final coalgebra structure. By viewing trees as formal power series, we develop a calculus where definitions are presented as behavioural differential equations. We present a general format for these equations that guarantees the existence and uniqueness of solutions. Although technically not very difficult, the resulting framework has surprisingly nice applications, which is illustrated by various concrete examples. 1
OverlapFree Symmetric D0L words
, 2001
"... Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlapfree infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the ThueMorse word w TM = 011010011001011010010110 ..."
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Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlapfree infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the ThueMorse word w TM = 01101001100101101001011001101001 : : :: It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlapfree words was studied e. g. by Fife [8] who described all binary overlapfree infinite words and Seebold [13] who proved that the ThueMorse word is essentially the only binary overlapfree word which is a fixed point of a morphism. Nowadays the theory of overlapfree words is a part of a more general theory of pattern avoidance [5]. J.P. Allouche and J. Shallit [2] asked if the initial Thue's construction of an overlapfree wo
A short proof of the transcendence of the Thue– Morse continued fraction
"... The ThueMorse sequence t = (tn)n≥0 on the alphabet {a, b} is defined as follows: tn = a (respectively, tn = b) if the sum of binary digits of n is even (respectively, odd). This famous binary sequence was first introduced by A. Thue [12] in 1912. It was considered nine years later by M. Morse [7] i ..."
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The ThueMorse sequence t = (tn)n≥0 on the alphabet {a, b} is defined as follows: tn = a (respectively, tn = b) if the sum of binary digits of n is even (respectively, odd). This famous binary sequence was first introduced by A. Thue [12] in 1912. It was considered nine years later by M. Morse [7] in a totally different context. These pioneering papers have led to a number of investigations and a broad literature devoted to t. There are many other ways to define the ThueMorse sequence. Each of them gives rise to specific interests, problems, and most of the time solutions. Such ubiquity is well described in the survey [1], where the occurrence of t in combinatorics, number theory, differential geometry, theoretical computer science, physics, and even chess is documented. For a and b distinct integers K. Mahler [6] (see also [2]) established that the sum of the series ∑ −n n≥0 tn2 is transcendental. The present note adresses another Diophantine result related to the ThueMorse sequence. It is widely believed that the continued fraction expansion of every irrational algebraic number α either is eventually periodic (and we know from Lagange’s theorem that this is the case if and only if α is a quadratic irrational) or contains arbitrarily large partial quotients. Apparently, this challenging question was first considered by A. Ya. Khintchin in [4] (see also [5], [11], or [13] for surveys or books including discussions of this subject). A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions with bounded partial quotients. In this direction, M. Queffélec [8] showed in 1998 that the ThueMorse continued fractions are transcendental. Theorem 1 (Queffélec). If a and b are distinct positive integers and t = (tn)n≥0 is the ThueMorse sequence on the alphabet {a, b}, then the number is transcendental. 1 ξ = [t0, t1, t2,...] = t0 +
PERIODIC UNIQUE BETAEXPANSIONS: THE SHARKOVSKIĬ ORDERING
"... ABSTRACT. Let β ∈ (1, 2). Each x ∈ [0, β−1] can be represented in the form x = εkβ −k, k=1 where εk ∈ {0, 1} for all k (a βexpansion of x). If β> 1+√5 2, then, as is well known, there 1 always exist x ∈ (0, β−1) which have a unique βexpansion. In the present paper we study (purely) periodic unique ..."
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ABSTRACT. Let β ∈ (1, 2). Each x ∈ [0, β−1] can be represented in the form x = εkβ −k, k=1 where εk ∈ {0, 1} for all k (a βexpansion of x). If β> 1+√5 2, then, as is well known, there 1 always exist x ∈ (0, β−1) which have a unique βexpansion. In the present paper we study (purely) periodic unique βexpansions and show that for each n ≥ 2 there exists βn ∈ [ 1+ √ 5 2, 2) such that there are no unique periodic βexpansions of smallest period n for β ≤ βn and at least one such expansion for β> βn. Furthermore, we prove that βk < βm if and only if k is less than m in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps. 1. HISTORY OF THE PROBLEM AND FORMULATION OF RESULTS This paper continues the line of research related to the combinatorics of representations of real numbers in noninteger bases ([12, 13, 15, 21]). 1 Let β ∈ (1, 2) be our parameter and let x ∈ Iβ: = [0,]. Then x has at least one represenβ−1 tation of the form (1.1) x = πβ(ε1, ε2,...): = εkβ −k, εk ∈ {0, 1}, k=1 (use, e.g., the greedy algorithm) which we call a βexpansion of x and write x ∼ (ε1, ε2,...)β. Let us recall some key results regarding βexpansions. Firstly, if 1 < β < G: = 1+√5, then 2 each x ∈ � � 1 0, has a continuum of βexpansions [12]. On the other hand, for any β> G, β−1 there exist infinitely many x which have a unique βexpansion (see [9, 13]), although almost all x ∈ Iβ still have a continuum of βexpansions [21]. More specifically, put x ∼ (010101...)β = 1 β2. Then both x and βx have a unique β
THE SINGULAR CONTINUOUS DIFFRACTION MEASURE OF THE THUEMORSE CHAIN
, 809
"... Abstract. The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the ThueMorse chain, in its realisation as a binary sequence with values in {±1}. We revisit this example and derive a functional equation together with an explicit form of the ..."
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Abstract. The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the ThueMorse chain, in its realisation as a binary sequence with values in {±1}. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product. The ThueMorse chain can be defined via the primitive substitution rule
Squarefree partial words
, 2008
"... We say that a partial word w over an alphabet A is squarefree if every factor xx ′ of w such that x and x ′ are compatible is either of the form ⋄a or a ⋄ where ⋄ is a hole and a ∈ A. We prove that there exist uncountably many squarefree partial words over a ternary alphabet with an infinite numbe ..."
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We say that a partial word w over an alphabet A is squarefree if every factor xx ′ of w such that x and x ′ are compatible is either of the form ⋄a or a ⋄ where ⋄ is a hole and a ∈ A. We prove that there exist uncountably many squarefree partial words over a ternary alphabet with an infinite number of holes.