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.

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 b-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit. 1.

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 non-quadratic 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

ABSTRACT. — It is well-known 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 Kodaira-Spencer 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 Kodaira-Spencer de cette courbe impliquent des bornes pour les exposants d’approximation diophantienne de la série,

Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the b-ary 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 g-ary 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 g-ary 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 g-ary 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

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]).

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 fixed-point D � Z2 and floating-point 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 well-known 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.

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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

Abstract. In contrast to Roth’s theorem that all algebraic irrational real numbers have approximation exponent two, the distribution of the exponents for the function field counterparts is not even conjecturally understood. We describe some recent progress made on this issue. An explicit continued fraction is not known even for a single non-quadratic algebraic real number. We provide many families of explicit continued fractions, equations and exponents for non-quadratic algebraic laurent series in finite characteristic, including non-Riccati examples with both bounded or unbounded sequences of partial quotients. On this occasion of Professor Abhyankar’s 70th birthday conference, it might be appropriate to mention some recent applications of the ‘high school algebra ’ [A] to the study of diophantine approximation for function fields in finite characteristic. This study is related to some of his loves: power series, continued fractions, algebraic curves, finite characteristic, resultants (and even automata). 1 What we know and don’t know about the basic