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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)

This paper deals with rational base number systems for p-adic numbers. We mainly focus on the system proposed by Akiyama et al. in 2008, but we also show that this system is in some sense isomorphic to some other rational base number systems by means of finite transducers. We identify the numbers with finite and eventually periodic representations and we… (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)

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)

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)

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)

Univoque numbers are real numbers λ > 1 such that the number 1 admits a unique expansion in base λ, i.e., a unique expansion 1 = j≥0 a j λ −(j+1) , with a j ∈ {0, 1,. .. , ⌈λ⌉ − 1} for every j ≥ 0. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called admissible sequences. We show how a 1983 study of the… (More)