5 research outputs found
Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers
We study arithmetical and combinatorial properties of -integers for
being the root of the equation . We determine with the accuracy of the maximal number of
-fractional positions, which may arise as a result of addition of two
-integers. For the infinite word coding distances between
consecutive -integers, we determine precisely also the balance. The word
is the fixed point of the morphism and . In the case the corresponding infinite word is
sturmian and therefore 1-balanced. On the simplest non-sturmian example with
, we illustrate how closely the balance and arithmetical properties of
-integers are related.Comment: 15 page
Abelian Complexity of Infinite Words Associated with Quadratic Parry Numbers
We derive an explicit formula for the Abelian complexity of infinite words
associated with quadratic Parry numbers.Comment: 12 page
Integers in number systems with positive and negative quadratic Pisot base
We consider numeration systems with base and , for quadratic
Pisot numbers and focus on comparing the combinatorial structure of the
sets and of numbers with integer expansion in base
, resp. . Our main result is the comparison of languages of
infinite words and coding the ordering of distances
between consecutive - and -integers. It turns out that for a
class of roots of , the languages coincide, while for other
quadratic Pisot numbers the language of can be identified only with
the language of a morphic image of . We also study the group
structure of -integers.Comment: 19 pages, 5 figure
Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers
In this paper we will deal with the balance properties of the infinite binary words associated to β-integers when β is a quadratic simple Pisot number. Those words are the fixed points of the morphisms of the type , for , , , where . We will prove that such word is t-balanced with . Finally, in the case that p < q it is known [B. Adamczewski, Theoret. Comput. Sci. 273 (2002) 197–224] that the fixed point of the substitution , is not m-balanced for any m. We exhibit an infinite sequence of pairs of words with the unbalance property