Algebraic Properties of Weak Perron Numbers

Counting Perron Numbers by Absolute Value

We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston [Thu].

Rényi β - expansions of 1 with β > 1 a real algebraic number , Perron numbers and a classification problem

We prove that for all algebraic number β > 1 the strings of zeros in the Rényi βexpansion dβ(1) of 1 exhibit a lacunarity bounded above by log(s(Pβ))/ log(β), where s(Pβ) is the size of the minimal polynomial of β. The conjecture about the specification of the β-shift, equivalently the uniform discreteness of the sets Zβ of β-integers, for β a Perron number is discussed. We propose a classifica...

On the dichotomy of Perron numbers and beta-conjugates

Let β > 1 be an algebraic number. A general definition of a beta-conjugate of β is proposed with respect to the analytical function fβ(z) = −1 + ∑ i≥1 tiz i associated with the Rényi β-expansion dβ(1) = 0.t1t2 . . . of unity. From Szegö’s Theorem, we study the dichotomy problem for fβ(z), in particular for β a Perron number: whether it is a rational fraction or admits the unit circle as natural...

The smallest Perron numbers

A Perron number is a real algebraic integer α of degree d ≥ 2, whose conjugates are αi, such that α > max2≤i≤d |αi|. In this paper we compute the smallest Perron numbers of degree d ≤ 24 and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. The computations use a family of explicit auxiliar...

Spectrum Properties of the Koopman and Frobenius-Perron Operators