Generic point equivalence and Pisot numbers

Pisot 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...

Pisot Numbers and Greedy Algorithm

Numbers , 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...

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.

Beta-representations of 0 and Pisot numbers

Let β > 1, d a positive integer, and Zβ,d = {z1z2 · · · | ∑ i>1 ziβ −i = 0, zi ∈ {−d, . . . , d}} be the set of infinite words having value 0 in base β on the alphabet {−d, . . . , d}. Based on a recent result of Feng on spectra of numbers, we prove that if the set Zβ,⌈β⌉−1 is recognizable by a finite Büchi automaton then β is a Pisot number. As a consequence of previous results, the set Zβ,d i...