Substitutions, abstract number systems and the space filling property

Fuchs, Clemens; Tijdeman, R.

doi:10.5802/aif.2243

Cited by 9 publications

(7 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the Tribonacci case they found a close connection with number systems. The latter connection was generalized by Fuchs and Tĳdeman in [3]. In the present paper we generalize the former property to unimodular substitutions defined over two letters.…”

Section: Introductionmentioning

confidence: 62%

Substitutions on two letters, cutting segments and their projections

Rosema¹

2007

J. Théor. Nombres Bordeaux

View full text Add to dashboard Cite

Substitutions on two letters, cutting segments and their projections par Sierk W. ROSEMA Résumé. Dans cet article on considère la structure des projections des segments de coupure correspondant aux substitutions unimodulaires sur un alphabet binaire. On montre qu'une telle projection est un bloc de lettres si et seulement si la substitution est sturmienne. Une double application de ce procédé à une substitution de Christoffel donne la substitution originelle. On obtient ainsi une dualité sur l'ensemble des substitutions de Christoffel.

show abstract

Section: Introductionmentioning

confidence: 62%

Substitutions on two letters, cutting segments and their projections

Rosema¹

2007

J. Théor. Nombres Bordeaux

View full text Add to dashboard Cite

show abstract

“…Analogous statements appear in various frameworks, see e.g. (Akiyama 2000), (Arnoux, Berthé, and Siegel 2004), (Barge and Diamond 2002), , (Fernique 2006), (Fuchs and Tijdeman 2006) or .…”

Section: The Ancestor Graphmentioning

confidence: 86%

“…It is expressed in (Berthé and Siegel 2005) in terms of the DumontThomas numeration. It also appears in (Fuchs and Tijdeman 2006) in a related context. Note that we can use the vast literature on the (F) property in the beta-numeration framework to exhibit classes of beta-substitutions that satisfy the geometric finiteness property (see e.g.…”

Section: Section 54mentioning

confidence: 94%

Substitutions, Rauzy fractals and tilings

Berthé

Siegel

Thuswaldner

2010

Combinatorics, Automata and Number Theory

View full text Add to dashboard Cite

“…, and by the induction hypothesis we obtain the result. Furthermore, to show the inclusion (10), it suffices to prove that ka ∈ αS[α], ∀ k ∈ Z, or equivalently (11) kb ∈ S[α], ∀k ∈ Z, as any rational integer may be written ε + ka for some ε ∈ S and k ∈ Z. Assume first that 0 < −b < a and choose S α = {0, .…”

Section: Proof Of Theorem 2 (Ii)mentioning

confidence: 99%

Comments on the height reducing property II

Akiyama¹,

Thuswaldner²,

Zaı̈mi³

2015

Indagationes Mathematicae

View full text Add to dashboard Cite

A complex number α is said to satisfy the height reducing property if there is a finite set F ⊂ Z such that Z[α] = F [α], where Z is the ring of the rational integers. It is easy to see that α is an algebraic number when it satisfies the height reducing property. We prove the relation Card(F ) ≥ max{2, |Mα(0)|}, where Mα is the minimal polynomial of α over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers α. In addition, we show that there is an algorithm to determine the minimal height polynomial of a given algebraic number, provided it has no conjugate of modulus one.Card(F ) ≤ Card(S)(Card(S) s+1 − 1)/(Card(S) − 1), and, hence, J (t−s) (0) = β which implies that β ∈ U . Thus U ⊂ ℘, a contradiction, because by (9) we have that cZ ⊂ U and so U cannot be finite.

show abstract

Substitutions, abstract number systems and the space filling property

Cited by 9 publications

References 34 publications

Substitutions on two letters, cutting segments and their projections

Substitutions on two letters, cutting segments and their projections

Substitutions, Rauzy fractals and tilings

Comments on the height reducing property II

Contact Info

Product

Resources

About