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Bull. London Math. Soc. 42 (2010) 538552 Ce2010 London Mathematical Society doi:10.1112/blms/bdq019

Summary: Bull. London Math. Soc. 42 (2010) 538­552 Ce2010 London Mathematical Society
Rational numbers with purely periodic -expansion
Boris Adamczewski, Christiane Frougny, Anne Siegel and Wolfgang Steiner
We study real numbers with the curious property that the -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 ()
denote the supremum of the real numbers c in (0, 1) such that all positive rational numbers less
than c have a purely periodic -expansion. We prove that () is irrational for a class of cubic
Pisot units that contains the smallest Pisot number . This result is motivated by the observation
of Akiyama and Scheicher that () = 0.666 666 666 086 . . . is surprisingly close to 2/3.
1. Introduction
One of the most basic results about decimal expansions is that every rational number has
an eventually periodic expansion (A sequence (an)n 1 is eventually periodic if there exists a
positive integer p such that an+p = an for every positive integer n large enough), the converse
being obviously true. In fact, much more is known for we can easily distinguish rationals with
a purely periodic expansion (A sequence (an)n 1 is purely periodic if there exists a positive


Source: Adamczewski, Boris - Institut Camille Jordan, Université Claude Bernard Lyon-I


Collections: Mathematics