Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers

Abstract

Concentration and equi-distribution, near the unit circle, in Solomyak's set, of the union of the Galois conjugates and the beta-conjugates of a Parry number $β$ are characterized by means of the Erdős-Turán approach, and its improvements by Mignotte and Amoroso, applied to the analytical function $f_β(z) = -1 + \sum_{i \geq 1} t_i z^i$ associated with the Rényi $β$-expansion $d_β(1)= 0.t_1 t_2 ...$ of unity. Mignotte's discrepancy function requires the knowledge of the factorization of the Parry polynomial of $β$. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaaler, Smyth, Schinzel in terms of cyclotomic, reciprocal non-cyclotomic and non-reciprocal factors. An upper bound of Mignotte's discrepancy function which arises from the beta-conjugates of $β$ which are roots of cyclotomic factors is linked to the Riemann hypothesis, following Amoroso. An equidistribution limit theorem, following Bilu's theorem, is formulated for the concentration phenomenon of conjugates of Parry numbers near the unit circle. Parry numbers are Perron numbers. Open problems on non-Parry Perron numbers are mentioned in the context of the existence of non-unique factorizations of elements of number fields into irreducible Perron numbers (Lind).