We survey the properties of sets of integers recognizable by automata when they are written in p-ary 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 Cobham-Semenov, the original proof being published in Russian. 1

We discuss a well-known binary sequence called the Thue-Morse sequence, or the Prouhet-Thue-Morse 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 Prouhet-Thue-Morse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.

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.

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.

The effective fractal dimensions at the polynomial-space level and above can all be equivalently defined as the C-entropy rate where C is the class of languages corresponding to the level of effectivization. For example, pspace-dimension is equivalent to the PSPACE-entropy rate. At lower levels of complexity the equivalence proofs break down. In the polynomialtime case, the P-entropy rate is a lower bound on the p-dimension. Equality seems unlikely, but separating the P-entropy rate from p-dimension would require proving P != NP. We show that at the finite-state level, the opposite of the polynomial-time case happens: the REG-entropy rate is an upper bound on the finite-state dimension. We also use the finitestate genericity of Ambos-Spies and Busse (2003) to separate finite-state dimension from the REG-entropy rate. However, we point out that a block-entropy rate characterization of finite-state dimension follows from the work of Ziv and Lempel (1978) on finite-state compressibility and the compressibility characterization of finite-state dimension by Dai, Lathrop, Lutz, and Mayordomo (2004). As applications of the REG-entropy rate upper bound and the block-entropy rate characterization, we prove that every regular language has finite-state dimension 0 and that normality is equivalent to finite-state dimension 1.

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, p-Lucas sequences, Chomsky-Schutzenberger 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 Chomsky-Schutzenberger theorem [16] asserts that the generating seri...

In recent years, there has been a number of papers about the combinatorial notion of symbolic complexity: this is the function counting the number of factors of length n for a sequence. The complexity is an indication of the degree of randomness of the sequence: a periodic sequence has a bounded complexity, the expansion of a normal number has an exponential complexity. For a given sequence, the complexity function is generally not of easy access, and it is a rich and instructive work to compute it; a survey of this kind of results can be found in [ALL]. We are interested here in further results in the theory of symbolic complexity, somewhat beyond the simple question of computing the complexity of various sequences. These lie mainly in two directions; first, we give a survey of an open question which is still very much in progress, namely: to determine which functions can be the symbolic complexity function of a sequence. Then, we investigate the links between the complexity of a sequence and its associated dynamical system, and insist on the cases where the knowledge of

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.

Let t be the infinite fixed point, starting with 1, of the morphism :0#01, 1 # 10. An infinite word over {0, 1} is said to be overlap-free if it contains no factor of the form axaxa, where a #{0,1}and x #{0,1} # . We prove that the lexicographically least infinite overlap-free binary word beginning with any specified prefix, if it exists, has a su#x which is a su#x of t. In particular, the lexicographically least infinite overlap-free binary word is 001001t. Keywords: Homomorphism, fixed point, overlap-free word. 1991 Mathematics Subject Classification: Primary 68R15. 1 the electronic journal of combinatorics 5 (1998),#R27 2 1 Introduction Since the pioneering work of Thue [14, 15] (see also [5]) the overlap-free words on a finite alphabet, i.e., those words that do not contain a factor axaxa,wherexis a word and a a letter, have been studied extensively. The question of extremality (for the lexicographic order) of overlap-free binary infinite words seems to have been addre...

Let b ≥ 2 be an integer. Émile Borel [9] conjectured that every real irrational algebraic number α should satisfy some of the laws shared by almost all real numbers with respect to their b-adic expansions. Despite some recent progress [1, 3, 7], we are still very far away from establishing such a strong result. In the present work, we are concerned