Charles Pisot

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.

Proximality in Pisot Tiling Spaces

A substitution φ is strong Pisot if its abelianization matrix is non-singular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space Tφ has pure discrete spectrum, we describe the collection T P φ of pairs of proximal tilings in Tφ in a natural way as a s...

The continuous Skolem-Pisot problem

We study decidability and complexity questions related to a continuous analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of a linear recurrent sequence. In particular, we show that the continuous version of the nonnegativity problem is NP-hard in general and we show that the presence of a zero is decidable for several subcases, including instances of depth two or less,...

On the Pisot substitution conjecture

Pisot Numbers and Greedy Algorithm