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Numeration systems, the basis of which is defined by a linear recurrence with integer coefficients, are considered. We give conditions on the recurrence under which the function of normalization which transforms any representation of an integer into the normal one—obtained by the usual algorithm—can be realized by a finite automaton. Addition is a(More)
A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique such expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite(More)
Every positive integer can be written as a sum of Fibonacci numbers; it can also be written as a ((nite) sum of (positive and negative) powers of the golden mean '. We show that there exists a letter-to-letter nite two-tape automaton that maps the Fibonacci representation of any positive integer onto its '-expansion, provided the latter is folded around the(More)
We study expansions in non-integer negative base −β introduced by Ito and Sadahiro [7]. Using countable automata associated with (−β)-expansions, we characterize the case where the (−β)-shift is a system of finite type. We prove that, if β is a Pisot number, then the (−β)-shift is a sofic system. In that case, addition (and more generally normalization on(More)
Let U be a strictly increasing sequence of integers. By a greedy algorithm, every nonnegative integer has a greedy U-representation. The successor function maps the greedy U-representation of N onto the greedy U-representation of N +1. We characterize the sequences U such that the successor function associated to U is a left, resp. a right sequential(More)
For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base 2. In this paper, we consider numeration systems with respect(More)
A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique such expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite(More)