Purely periodic β-expansions in the Pisot non-unit case
نویسندگان
چکیده
منابع مشابه
Purely Periodic Β-expansions in the Pisot Non-unit Case
It is well known that real numbers with a purely periodic decimal expansion are rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to betaexpansions with a Pisot base beta which is not necessarily a unit. We characterize real numbers having a purely periodic expansion in such a base. This characterization is given in terms of an explici...
متن کاملPurely Periodic β-Expansions with Pisot Unit Bases over Laurent Series
The present paper deals with β-expansions in algebraic function fields. If β is a Pisot unit, we characterise the elements whose β-expansion is purely periodic. In order to pursue this characterisation, we introduce a variant of the Rauzy fractal.
متن کاملBoundary of Central Tiles Associated with Pisot Beta-numeration and Purely Periodic Expansions
This paper studies tilings related to the β-transformation when β is a Pisot number (that is not supposed to be a unit). Then it applies the obtained results to study the set of rational numbers having a purely periodic β-expansion. Secial focus is given to some quadratic examples.
متن کاملΒ-expansions with Deleted Digits for Pisot Numbers Β
An algorithm is given for computing the Hausdorff dimension of the set(s) Λ = Λ(β,D) of real numbers with representations x = ∑∞ n=1 dnβ −n, where each dn ∈ D, a finite set of “digits”, and β > 0 is a Pisot number. The Hausdorff dimension is shown to be log λ/ log β, where λ is the top eigenvalue of a finite 0-1 matrix A, and a simple algorithm for generating A from the data β,D is given.
متن کاملRational 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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2007
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2007.07.005