منابع مشابه
On univoque Pisot numbers
We study Pisot numbers β ∈ (1, 2) which are univoque, i.e., such that there exists only one representation of 1 as 1 = P n≥1 snβ −n, with sn ∈ {0, 1}. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.
متن کاملOn certain computations of Pisot numbers
This paper presents two algorithms on certain computations about Pisot numbers. Firstly, we develop an algorithm that finds a Pisot number α such that Q[α] = F given a real Galois extension F of Q by its integral basis. This algorithm is based on the lattice reduction, and it runs in time polynomial in the size of the integral basis. Next, we show that for a fixed Pisot number α, one can comput...
متن کاملPisot Numbers in the Neighborhood of a Limit Point . II
Let S denote the set of real algebraic integers greater than one, all of whose other conjugates lie within the unit circle. In an earlier paper, we introduced the notion of "width" of a limit point a of S and showed that, if the width of a is smaller than 1.28... then there is an algorithm for determining all members of S in a neighborhood of a. Recently, we introduced the "derived tree" in ord...
متن کاملOn beta expansions for Pisot numbers
Given a number β > 1, the beta-transformation T = Tβ is defined for x ∈ [0, 1] by Tx := βx (mod 1). The number β is said to be a betanumber if the orbit {Tn(1)} is finite, hence eventually periodic. In this case β is the root of a monic polynomial R(x) with integer coefficients called the characteristic polynomial of β. If P (x) is the minimal polynomial of β, then R(x) = P (x)Q(x) for some pol...
متن کاملNumbers , Pisot Numbers , Mahler Measure and Graphs
We use graphs to define sets of Salem and Pisot numbers, and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers n the smallest known element of the n-th derived set of the set of Pisot numbers comes from a graph. We define the Mahler measur...
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ژورنال
عنوان ژورنال: Journal de Théorie des Nombres de Bordeaux
سال: 2004
ISSN: 1246-7405
DOI: 10.5802/jtnb.446