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16
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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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.
Diophantine properties of real numbers generated by finite automata
 Compos. Math
"... Abstract. We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the badic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit. 1. ..."
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Cited by 10 (3 self)
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Abstract. We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the badic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit. 1.
Diophantine approximation exponents and continued fractions for algebraic power series
 J. Number Theory
, 1999
"... 1998 For each rational number not less than 2, we provide an explicit family of continued fractions of algebraic power series in finite characteristic (together with the algebraic equations they satisfy) which has that rational number as its diophantine approximation exponent. We also provide some n ..."
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Cited by 8 (2 self)
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1998 For each rational number not less than 2, we provide an explicit family of continued fractions of algebraic power series in finite characteristic (together with the algebraic equations they satisfy) which has that rational number as its diophantine approximation exponent. We also provide some nonquadratic examples with bounded sequences of partial quotients. 1999 Academic Press Continued fraction expansions of real numbers and laurent series over finite fields are well studied because, for example, of their close connection with best diophantine approximations. In both the cases, the expansion terminates exactly for rationals and is eventually periodic exactly for quadratic irrationals. But continued fraction expansion is not known even for a single algebraic real number of degree more than two. It is not even known whether the sequence of partial quotients is bounded or not for such a number. (Because of the numerical evidence and a belief that algebraic numbers are like most numbers in this respect, it is often conjectured that the sequence is unbounded.) It is hard to obtain such expansions for algebraic numbers, because the effect of basic algebraic operations (except for adding an integer or, more generally, an integral Mobius transformation of determinant \1), such as addition or multiplication or even multiple or power, is not at all transparent on the continued fraction expansions. In finite characteristic p, the algebraic operation of taking pth power has a very transparent effect: If:=[a0, a1,...], then: p =[a p p
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]).
WORDS AND TRANSCENDENCE
, 2009
"... Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the bary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the gary expansion of x should sati ..."
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Cited by 5 (2 self)
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Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the bary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the gary expansion of x should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. We are very far from such a goal: there is no explicitly known example of a triple (g, a, x), where g � 3 is an integer, a a digit in {0,...,g − 1} and x a real irrational algebraic number, for which one can claim that the digit a occurs infinitely often in the gary expansion of x. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress has been made recently, thanks mainly to clever use of Schmidt’s subspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first [28] published in 1909 and the second [29] in 1950, Borel studied the gary expansion of real numbers, where g � 2 is a positive integer. In his second paper, he suggested that this expansion for a real irrational algebraic number should satisfy some of the laws shared by almost all numbers, in the sense of Lebesgue measure. Let g � 2 be an integer. Any real number x has a unique expansion x = a−kg k + ···+ a−1g + a0 + a1g −1 + a2g −2 + ·· ·, where k � 0 is an integer and the ai for i � −k, namely the digits of x in the expansion in base g of x, belong to the set {0, 1,...,g − 1}. Uniqueness is subject to the condition that the sequence (ai)i�−k is not ultimately constant and equal to g − 1. We write this expansion x = a−k ···a−1a0.a1a2 ·· ·. in base 10 (decimal expansion), whereas
Diophantine approximation and deformation
 Bull. Soc. Math. France
"... ABSTRACT. — It is wellknown that while the analogue of Liouville’s theorem on diophantine approximation holds in finite characteristic, the analogue of Roth’s theorem fails quite badly. We associate certain curves over function fields to given algebraic power series and show that bounds on the rank ..."
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ABSTRACT. — It is wellknown that while the analogue of Liouville’s theorem on diophantine approximation holds in finite characteristic, the analogue of Roth’s theorem fails quite badly. We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of KodairaSpencer map of this curves imply bounds on the diophantine approximation exponents of the power series, with more ‘generic ’ curves (in the deformation sense) giving lower exponents. If we transport Vojta’s conjecture on height inequality to finite characteristic by modifying it by adding suitable deformation theoretic condition, then we see that the numbers giving rise to general curves approach Roth’s bound. We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree. RÉSUMÉ.—APPROXIMATION DIOPHANTIENNE ET DÉFORMATION. — Alors que l’analogue du théorème de Liouville sur l’approximation diophantienne se conserve en caractéristique finie, il est bien connu que l’analogue du théorème de Roth échoue lamentablement. En associant àdesséries de puissances algébriques données certaines courbes sur les corps de fonctions, nous prouvons que des bornes pour le rang de l’application de KodairaSpencer de cette courbe impliquent des bornes pour les exposants d’approximation diophantienne de la série,
Continued fractions and transcendental numbers
"... It is widely believed that the continued fraction expansion of every irrational algebraic number α either is eventually periodic (and we know that this is the case if and only if α is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first con ..."
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Cited by 3 (3 self)
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It is widely believed that the continued fraction expansion of every irrational algebraic number α either is eventually periodic (and we know that this is the case if and only if α is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine in [22] (see also [6,39,41] for surveys including a
Digital algebra and circuits
 In LNCS 2772
, 2003
"... Abstract. Digital numbers D are the world’s most popular data representation: nearly all texts, sounds and images are coded somewhere in time and space by binary sequences. The mathematical construction of the fixedpoint D � Z2 and floatingpoint D ′ � Q2 digital numbers is a dual to the classical ..."
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Cited by 1 (0 self)
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Abstract. Digital numbers D are the world’s most popular data representation: nearly all texts, sounds and images are coded somewhere in time and space by binary sequences. The mathematical construction of the fixedpoint D � Z2 and floatingpoint D ′ � Q2 digital numbers is a dual to the classical constructions of the real numbers R. The domain D ′ contains the binary integers N and Z, as well as Q. The arithmetic operations in D ′ are the usual ones when restricted to integers or rational numbers. Similarly, the polynomial operations in D ′ are the usual ones when applied to finite binary polynomials F2[z] or their quotients F2(z). Finally, the set operations in D ′ are the usual ones over finite or infinite subsets of N. The resulting algebraic structure is rich, and we identify over a dozen rings, fields and Boolean algebras in D ′. Each structure is wellknown in its own right. The unique nature of D ′ is to combine all into a single algebraic structure, where operations of different nature happily mix.
Transcendental Continued Fractions over IK p(X)
"... The purpose behind this work is to construct from a family of algebraic formal power series of degree more than 2, a family of transcendental fractions over IKp(X). 1 ..."
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The purpose behind this work is to construct from a family of algebraic formal power series of degree more than 2, a family of transcendental fractions over IKp(X). 1