Rational numbers with purely periodic β-expansion

We study real numbers beta with the curious property that the beta-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 previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let gamma(beta) denote the supremum of the real numbers c in (0, 1) such that all positive rational numbers less than c have a purely periodic beta-expansion. We prove that gamma(beta) is irrational for a class of cubic Pisot units that contains the smallest Pisot number eta. This result is motivated by the observation of Akiyama and Scheicher that gamma(eta) = 0.666 666 666 086 ... is surprisingly close to 2/3.