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MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000000
 

Summary: MATHEMATICS OF COMPUTATION
Volume 00, Number 0, Pages 000­000
S 0025-5718(XX)0000-0
ON UNIVOQUE PISOT NUMBERS
JEAN-PAUL ALLOUCHE, CHRISTIANE FROUGNY, AND KEVIN G. HARE
Abstract. We study Pisot numbers (1, 2) which are univoque, i.e., such
that there exists only one representation of 1 as 1 = n1 sn-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.
1. Introduction
Representations of real numbers in non-integer bases were introduced by R´enyi
[27] and first studied by R´enyi and by Parry [26, 27]. Among the questions that were
addressed is the uniqueness of representations. Given a sequence (sn)n1, Erdos,
Jo´o and Komornik, [20], gave a purely combinatorial characterization for when
there exists (1, 2) such that 1 = n1 sn-n
is the unique representation of 1.
This set of binary sequences is essentially the same as a set studied by Cosnard and
the first author [1, 2, 4] in the context of iterations of unimodal continuous maps
of the unit interval.

  

Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11

 

Collections: Computer Technologies and Information Sciences