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

A word on a finite alphabet A is said to be overlap-free if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlap-free binary words of length n, which is known to be bounded by a polynomial in n. First, we describe a bijection between the set of overlap-free words and a rational language. This yields recurrence relations for un , which allow to compute un in logarithmic time. Then, we prove that the numbers ff = sup f r j n r = O (un) g and fi = inf f r j un = O (n r ) g are distinct, and we give an upper bound for ff and a lower bound for fi. Finally, we compute an asymptotically tight bound to the number of overlap-free words of length less than n. 1 Introduction In general, the problem of evaluating the number un of words of length n in the language U consisting of words on some finite alphabet A with no factors in a certain set F is not easy. If F is finite, it amounts to counting words in...

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...

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.

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Dirichlet series whose coefficients are generated by finite automata define meromorphic functions on the whole complex plane. As consequences, a new proof of Cobham's theorem on the existence of logarithmic frequencies of symbols in automatic sequences is given, and certain infinite products are explicitly computed. Keywords: automatic sequences, Dirichlet series, logarithmic frequency. AMS classification: 11B85, 11M41, 68R15. 1 Introduction Automatic sequences have many properties ranging from number theory to harmonic analysis, from theoretical computer science to physics. See for example [11], [2], or [5]. An intuitive definition is that, given an integer d 2, a sequence (u n ) n0 with values in a finite set is d-automatic if its n-th term can be computed by a finite-state machine using the base d expansion of the integer n. (A precise definition is given below.) In his seminal paper on automatic sequences [10], Cobham proves that, given an automatic sequence (u n ) n0 , the se...

s(n) + s(n + 1). Stern showed in 1858 that gcd(s(n), s(n + 1)) = 1, and that every positive rational number a s(n) b occurs exactly once in the form s(n+1) for some n ≥ 1. We show that in a strong sense, the average value of these fractions is 3 2. We also show that for d ≥ 2, the pair (s(n), s(n+1)) is uniformly distributed among all feasible pairs of congruence classes modulo d. More precise results are presented for d = 2 and 3. 1

We discuss the summation of certain series defined by counting blocks of digits in the B-ary expansion of an integer. For example, if s2(n) denotes the sum of the base-2 digits of n, we show that � n≥1 s2(n)/(2n(2n + 1)) = (γ + log 4 π)/2. We recover this previous result of Sondow and provide several generalizations. MSC: 11A63, 11Y60. 1

Stern polynomials B k (t), k R, are introduced in the following way: B 0 (t) = 0, B 1 (t) = 1, B 2n (t) = tBn (t), and B 2n+1 (t) = Bn+1 (t) + Bn (t).

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