 
Summary: MATHEMATICS OF COMPUTATION
Volume 00, Number 0, Pages 000000
S 00255718(XX)00000
ON UNIVOQUE PISOT NUMBERS
JEANPAUL 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 snn, 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 noninteger 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 snn
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.
