On univoque Pisot numbers

Abstract

We study Pisot numbers β ∈ ( 1 , 2 ) \beta \in (1, 2) which are univoque, i.e., such that there exists only one representation of 1 1 as 1 = ∑ n ≥ 1 s n β − n 1 = \sum _{n \geq 1} s_n\beta ^{-n} , with s n ∈ { 0 , 1 } s_n \in \{0, 1\} . We prove in particular that there exists a smallest univoque Pisot number, which has degree 14 14 . Furthermore we give the smallest limit point of the set of univoque Pisot numbers.