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

Abstract. We define a two-sided analog of Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on T 2 which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions. Among numerous connections between ergodic theory and the metric theory of numbers, the questions related to algebraic irrationals, expansions associated with them and ergodic properties of related dynamical systems, are of special interest. The simplest case, i.e. the golden ratio, the Fibonacci automorphism etc., has

A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a non-negative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is order-preserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an order-preserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1

We study the distributions F`;p of the random sums P 1 1 " n` n , where " 1 ; " 2 ; : : : are i.i.d. Bernoulli-p and ` is the inverse of a Pisot number (an algebraic integer fi whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = :5, F`;p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain fi of small degree, simulation gives the Hausdorff dimension to several decimal places.

Every positive integer can be written as a sum of Fibonacci numbers; it can also be written as a (finite) sum of (positive and negative) powers of the golden mean '. We show that there exists a letter-to-letter finite two-tape automaton that maps the Fibonacci representation of any positive integer onto its '-expansion, provided the latter is folded around the radix point. As a corollary, the set of '-expansions of the positive integers is a linear context-free language. These results are actually proved in the more general case of quadratic Pisot units. R'esum'e Tout nombre entier positif peut s"ecrire comme une somme de nombres de Fibonacci; tout entier peut 'egalement s"ecrire comme une somme (finie) de puissances (positives et n'egatives) du "nombre d'or" '. Nous montrons qu'il existe un automate `a deux bandes, fini et lettre-`a-lettre, qui envoie la repr'esentation d'un entier en base de Fibonacci sur sa repr'esentation dans la base ' modulo le fait qu'on a repli'e cette derni`e...

Abstract. We study the arithmetic codings of hyperbolic automorphisms of the 2-torus, i.e. the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of additive group onto the torus which preserve this structure and turn the two-sided shift into a given automorphism of the torus. This group is uniquely defined by an automorphism, and such an arithmetic coding is a homomorphism of that group onto T 2. The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtained; it is formulated in terms of a certain binary quadratic form constructed by means of a given automorphism. Furthermore, we describe all bijective arithmetic codings in terms the Dirichlet group of the corresponding quadratic field. The minimum of that quadratic form over the nonzero elements of the lattice coincides with the minimal possible order of the kernel of a homomorphism described above. In this work we continue studying the symbolic dynamics of ergodic automorphisms of the 2-torus. The dynamics of automorphisms of the torus is related more

Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.

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We consider numeration systems where the base is a negative integer, or a complex number which is a root of a negative integer. We give parallel algorithms for addition in these numeration systems, from which we derive on-line algorithms realized by finite automata. A general construction relating addition in base fi and addition in base fi m is given. Results on addition in base fi = m p b, where b is a relative integer, follow. We also show that addition in base the golden ratio is computable by an on-line finite automaton, but is not parallelizable. 1 Introduction A positional numeration system is given by a base and by a set of digits. In the most usual numeration systems, the base is an integer b 2 and the digit set is f0; : : : ; b \Gamma 1g. In order to represent complex numbers without separating the real and the imaginary part, one can use a complex base. For instance, it is known that every complex number can be expressed with base i p 2 and digit set f0; 1g (...

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