Cubic Pisot units with finite beta expansions

Beta Expansions for Regular Pisot Numbers

A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is, in general, infinitely long and non-repeating, it is known that if the base is a Pisot number, then this expansion will always be finite or periodic. Some work has been done to learn more about these expansions...

Voronoi Cells of Beta-Integers

In this paper are considered one-dimensional tilings arising from some Pisot numbers encountered in quasicrystallography as the quadratic Pisot units and the cubic Pisot unit associated with 7-fold symmetry, and also the Tribonacci number. We give characterizations of the Voronoi cells of such tilings, using word combinatorics and substitutions.

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit β and the greedy β-transformation. In this paper, we consider different transformations generating expansions in base β, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. W...

Minimal Weight Expansions in Some Pisot Bases

For numeration systems representing real numbers and integers, which are based on Pisot numbers, we study expansions with signed digits which are minimal with respect to the absolute sum of digits. It is proved that these expansions are recognizable by a finite automaton if the base β is the root of a polynomial whose (integer) coefficients satisfy a certain condition (D). When β is the Golden ...

beta-EXPANSIONS WITH DELETED DIGITS FOR PISOT NUMBERS beta