2006/03/28 by P. Ambroz, Ambroz, P., C. Frougny +1
Mathematics · #11A63 #11R06 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A63 #msc:11R06
paper · pdf · doi:10.48550/arxiv.math/0603650
20 pages
arxiv created 2006/03/28 · arxiv updated 2009/12/01
We study α-adic expansions of numbers in an extension field, that is to say, left infinite representations of numbers in the positional numeration system with the base α, where α is an algebraic conjugate of a Pisot number β. Based on a result of Bertrand and Schmidt, we prove that a number belongs to ℚ(α) if and only if it has an eventually periodic α-expansion. Then we consider α-adic expansions of elements of the extension ring ℤ[α-1] when β satisfies the so-called Finiteness property (F). In the particular case that β is a quadratic Pisot unit, we inspect the unicity and/or multiplicity of α-adic expansions of elements of ℤ[α-1]. We also provide algorithms to generate α-adic expansions of rational numbers.