Abstract
Analysis of Algorithms The β-numeration, born with the works of Rényi and Parry, provides a generalization of the notions of integers, decimal numbers and rational numbers by expanding real numbers in base β, where β>1 is not an integer. One of the main differences with the case of numeration in integral base is that the sets which play the role of integers, decimal numbers and rational numbers in base β are not stable under addition or multiplication. In particular, a fractional part may appear when one adds or multiplies two integers in base β. When β is a Pisot number, which corresponds to the most studied case, the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β are bounded by constants which only depend on β. We prove that, for any Perron number β, the set of finite or ultimately periodic fractional parts of the sum, or the product, of two integers in base β is finite. Additionally, we prove that it is possible to compute this set for the case of addition when β is a Parry number. As a consequence, we deduce that, when β is a Perron number, there exist bounds, which only depend on β, for the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β. Moreover, when β is a Parry number, the bound associated with the case of addition can be explicitly computed.
Highlights
The β-numeration, born in the late 50’s with the works of Renyi [39] and Parry [35], is a generalization of numeration in a non-integer base which enables a modelling of quasicrystals [43]
The number systems defined by the β-numeration are closely related to canonical number systems [6, 40, 5], number systems generated by iterated function systems [45] or by substitutive systems of Pisot type [20]
A common feature between these fields is the property of self-similarity, which yields results in number theory [7], geometry [29], topology [8, 19], dynamical systems [46, 37, 38], combinatorics on words [22] and theoretical computer science [9]
Summary
The β-numeration, born in the late 50’s with the works of Renyi [39] and Parry [35], is a generalization of numeration in a non-integer base which enables a modelling of quasicrystals [43]. In order to perform arithmetics on β-integers, say for instance to compute the addition of two β-integers, one must be able to renormalize expansions in base β of real numbers obtained after adding β-integers. Several examples are studied in [9], where a method is described in order to compute upper bounds of L⊕ and L⊙ for Pisot numbers satisfying additional algebraic properties. These geometrical representation of expansions are contained in a bounded subset of Rd, where d is the degree of β.
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