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35
Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
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Cited by 68 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
The ubiquitous ProuhetThueMorse sequence
 Sequences and their applications, Proceedings of SETA’98
, 1999
"... We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be som ..."
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Cited by 59 (9 self)
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We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.
The Ring of kRegular Sequences
, 1992
"... The automatic sequence is the central concept at the intersection of formal language theory and number theory. It was introduced by Cobham, and has been extensively studied by Christol, Kamae, Mendes France and Rauzy, and other writers. Since the range of automatic sequences is nite, however, their ..."
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Cited by 39 (7 self)
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The automatic sequence is the central concept at the intersection of formal language theory and number theory. It was introduced by Cobham, and has been extensively studied by Christol, Kamae, Mendes France and Rauzy, and other writers. Since the range of automatic sequences is nite, however, their descriptive power is severely limited.
Determination Of The Algebraic Relations Among Special Values In Positive Characteristic
 ANN. OF MATH
, 2003
"... We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual tmotives, we nd that all algebraic relations among special values of the geometricfunction over Fq [T ] are explained by the standard functional equations. ..."
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Cited by 36 (9 self)
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We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual tmotives, we nd that all algebraic relations among special values of the geometricfunction over Fq [T ] are explained by the standard functional equations.
On the complexity of algebraic numbers I. Expansions in integer bases
, 2005
"... Let b ≥ 2 be an integer. We prove that the badic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. O ..."
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Cited by 33 (21 self)
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Let b ≥ 2 be an integer. We prove that the badic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.
Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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Cited by 22 (3 self)
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
Transcendence of Formal Power Series With Rational Coefficients
, 1999
"... We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal po ..."
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Cited by 9 (2 self)
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We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal power series, binomial series, automatic sequences, pLucas sequences, ChomskySchutzenberger theorem. 1 Introduction Formal power series with integer coefficients often occur as generating series. Suppose that a set E contains exactly a n elements of "size" n for each integer n: the generating series of this set is the formal power series P n0 a n X n (this series belongs to Z[[X]] ae Q[[X]]). Properties of this formal power series reflect properties of its coefficients, and hence properties of the set E. Roughly speaking, algebraicity of the series over Q(X) means that the set has a strong structure. For example, the ChomskySchutzenberger theorem [16] asserts that the generating seri...
Real and padic Expansions Involving Symmetric Patterns
"... This paper is motivated by the nonArchimedean counterpart of a problem raised by Mahler and Mendès France, and by questions related to the expected normality of irrational algebraic numbers. We introduce a class of sequences enjoying a particular combinatorial property: the precocious occurrences o ..."
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Cited by 9 (8 self)
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This paper is motivated by the nonArchimedean counterpart of a problem raised by Mahler and Mendès France, and by questions related to the expected normality of irrational algebraic numbers. We introduce a class of sequences enjoying a particular combinatorial property: the precocious occurrences of infinitely many symmetric patterns. Then, we prove several transcendence statements involving both real and padic numbers associated with these palindromic sequences. 1
On Continued Fraction Expansions in Positive Characteristic: Equivalence Relations and Some Metric Properties
"... The aim of this paper is to survey some properties of analogues of continued fraction expansions for formal power series with coefficients in a finite field. We discuss in particular connections between equivalence relations for continued fractions and the action of SL(2; F q [X]). ..."
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Cited by 5 (2 self)
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The aim of this paper is to survey some properties of analogues of continued fraction expansions for formal power series with coefficients in a finite field. We discuss in particular connections between equivalence relations for continued fractions and the action of SL(2; F q [X]).