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 study the number of local configurations in a discrete plane. We convert this problem into a computation of a double sequence complexity. We compute the number C(n; m) of distinct n \Theta m patterns appearing in a discrete plane. We show that C(n; m) = nm for all n and m positive integers. The coding of this sequence by a Z 2 -action on the unidimensional torus gives information about the structure of a discrete plane. Furthermore, this sequence is a generalized Rote sequence with complexity P (n; m) = 2nm for all n and m positive integers and with a symmetric complementary language for rectangular words. Key-words: Discrete planes, double sequence complexity, plane partitions, symmetric complementary language, generalization of Rote and Sturmian sequences. 1 Introduction In this article, we use the notion of complexity for a double sequence to study local configurations in a discrete plane. The complexity function p(n) for a sequence in a finite alphabet is defined from N to...

Abstract. We present some asymptotic results about the frequency of a letter appearing in a generalized unidimensional automatic sequence. Next, we study multidimensional generalized automatic sequences and the corresponding frequencies. 1.

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We introduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a Z²-action by rotations on the circle. This gives a new geometric interpretation of the Jacobi-Perron algorithm, as a map operating on the parameter space of Z²-actions by rotations.

journal homepage: www.elsevier.com/locate/disc Multidimensional generalized automatic sequences and