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32
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 66 (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 55 (8 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.
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.
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...
Sequences Of Low Complexity: Automatic And Sturmian Sequences
"... The complexity function is a classical measure of disorder for sequences with values in a finite alphabet: this function counts the number of factors of given length. We introduce here two characteristic families of sequences of low complexity function: automatic sequences and Sturmian sequences ..."
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Cited by 6 (0 self)
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The complexity function is a classical measure of disorder for sequences with values in a finite alphabet: this function counts the number of factors of given length. We introduce here two characteristic families of sequences of low complexity function: automatic sequences and Sturmian sequences. We discuss their topological and measuretheoretic properties, by introducing some classical tools in combinatorics on words and in the study of symbolic dynamical systems. 1 Introduction The aim of this course is to introduce two characteristic families of sequences of low "complexity": automatic sequences and Sturmian sequences (complexity is defined here as the combinatorial function which counts the number of factors of given length of a sequence over a finite alphabet). These sequences not only occur in many mathematical fields but also in various domains as theoretical computer science, biology, physics, cristallography... We first define some classical tools in combinatorics on...
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 6 (5 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]).
Computational Classification of Numbers and Algebraic Properties
"... In this paper, we propose a computational classification of finite characteristic numbers ..."
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Cited by 4 (2 self)
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In this paper, we propose a computational classification of finite characteristic numbers
(NON)AUTOMATICITY OF NUMBER THEORETIC FUNCTIONS
, 2008
"... Denote by λ(n) Liouville’s function concerning the parity of the number of prime divisors of n. Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that λ(n) is not k–automatic for any k> 2. This yields ..."
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Cited by 3 (1 self)
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Denote by λ(n) Liouville’s function concerning the parity of the number of prime divisors of n. Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that λ(n) is not k–automatic for any k> 2. This yields that P∞ n=1 λ(n)Xn ∈ Fp[[X]] is transcendental over Fp(X) for any prime p> 2. Similar results are proven (or reproven) for many common number–theoretic functions, including ϕ, µ, Ω, ω, ρ, and others.