Periodic expansion of one by Salem numbers

On the beta expansion for Salem numbers of degree 6

For a given β > 1, the beta transformation T = Tβ is defined for x ∈ [0, 1] by Tx := βx (mod 1). The number β is said to be a beta number if the orbit {T(1)}n≥1 is finite, hence eventually periodic. It is known that all Pisot numbers are beta numbers, and it is conjectured that this is true for Salem numbers, but this is known only for Salem numbers of degree 4. Here we consider some computatio...

Small Salem Numbers

Salem numbers of negative trace

We prove that, for all d ≥ 4, there are Salem numbers of degree 2d and trace −1, and that the number of such Salem numbers is d/ (log log d). As a consequence, it follows that the number of totally positive algebraic integers of degree d and trace 2d − 1 is also d/ (log log d).

RATIONAL NUMBERS WITH PURELY PERIODIC β-EXPANSION by

— We study real numbers β with the curious property that the β-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previou...

Rational numbers with purely periodic β - expansion

We study real numbers β with the curious property that the β-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previousl...