Bertrand numeration systems and recognizability

Bruyère, Véronique; Hansel, Georges

doi:10.1016/s0304-3975(96)00260-5

Cited by 60 publications

(83 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The integer val(w) is said to be the numerical value of w. So each non-negative integer n is represented by a unique word val −1 (n) ∈ L and this leads to the notion of numeration system on a regular language. These systems have been introduced in [22] and generalize classical numeration systems like the k-adic systems, the Fibonacci system and the linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number (for the properties of these latter systems we refer to [6]). …”

Section: Preliminariesmentioning

confidence: 99%

Distribution of Additive Functions with Respect to Numeration Systems on Regular Languages

Grabner

Rigo

2006

Theory Comput Syst

View full text Add to dashboard Cite

show abstract

Section: Preliminariesmentioning

confidence: 99%

Distribution of Additive Functions with Respect to Numeration Systems on Regular Languages

Grabner

Rigo

2006

Theory Comput Syst

View full text Add to dashboard Cite

show abstract

“…If w is the nth word of the genealogically ordered language L for some n ∈ N (positions inside L are counted from 0), then we write val(w) = n and we say that w is the representation of n or that n is the numerical value of w (the abstract numeration system S being understood). This way of representing nonnegative integers has been first introduced in [7] and generalizes classical numeration systems like the positional systems built over linear recurrent sequences of integers whose characteristic polynomial is the minimal polynomial of a Pisot number [2].…”

Section: Preliminariesmentioning

confidence: 99%

Abstract \beta -expansions and ultimately periodic representations

Rigo¹,

Steiner²

2005

J. Théor. Nombres Bordeaux

View full text Add to dashboard Cite

L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb.cedram. org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Abstract. For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is Q(β) if the dominating eigenvalue β > 1 of the automaton accepting the language is a Pisot number. Moreover, if β is neither a Pisot nor a Salem number, then there exist points in Q(β) which do not have any ultimately periodic representation. IntroductionIn [7], abstract numeration systems on regular languages have been introduced. They generalize in a natural way a large variety of classical positional systems like the q-ary system or the Fibonacci system: each nonnegative integer n is represented by the nth word of an ordered infinite regular language L. For instance, considering the natural ordering of the digits, the genealogical enumeration of the words belonging to the language L = {0}∪{1, . . . , q−1}{0, . . . , q−1} * (resp. L = {0, 1}∪{10}{10, 0} * {λ, 1}) leads back to the q-ary (resp. the Fibonacci) system. Later on, this setting has been extended to allow the representation of real numbers as well as of integers.Various notions appearing in number theory, in formal languages theory or in the analysis of algorithms depend on how numbers are represented.The second author was supported by the Austrian Science Foundation FWF, grant S8302-MAT. 284Michel Rigo, Wolfgang Steiner So these abstract systems have led to new nontrivial applications. To cite just a few: the characterization of the so-called recognizable sets of integers, the investigation of the dynamical and topological properties of the "odometers" or the study of the asymptotic behavior of the corresponding "sum-of-digits" function.As we will see, it turns out that this way of representing real numbers is a quite natural generalization of Rényi's β-expansions [11]. More precisely, the primitive automata lead to the representations of real numbers based on substitutions introduced by Dumont and Thomas [3]. The nonprimitive automata provide new numeration systems.Real numbers having an ultimately periodic representation deserve a special attention. Indeed, for the q-ary system, these numbers are exactly the rational numbers. More generally, the set of ultimately periodic representations is dense in the set of all the admissible representations and therefore this rises number-theoretic questions like the approximation of real numbers by numbers having ultimately periodic expansions.On the one hand, for Rényi's classical β-expansions it is well-known that t...

show abstract

“…In contrast, the famous theorem of Cobham [7] states that the only sets of natural numbers that are both p-and qrecognizable, when p and q are two multiplicatively independent integers > 1, are unions of arithmetic progressions, and thus are k-recognizable for any integer k > 1. Several generalizations of Cobham's Theorem have been given, see for instance [25,10,6,17,8]. In particular this result has been extended by Bès [4] to non-standard numeration systems.…”

Section: Introductionmentioning

confidence: 99%

“…This implies that a set of integers which is U -recognizable is then also Y -recognizable. Note that in [6] it is proved that if U and V are two linear numeration systems with the same characteristic polynomial which is the minimal polynomial of a Pisot number, then a U -recognizable set is also V -recognizable.…”

Section: Introductionmentioning

confidence: 99%

On multiplicatively dependent linear numeration systems, and periodic points

Frougny

2002

RAIRO-Theor. Inf. Appl.

View full text Add to dashboard Cite

Abstract. Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.

show abstract

Bertrand numeration systems and recognizability

Cited by 60 publications

References 12 publications

Distribution of Additive Functions with Respect to Numeration Systems on Regular Languages

Distribution of Additive Functions with Respect to Numeration Systems on Regular Languages

Abstract \beta -expansions and ultimately periodic representations

On multiplicatively dependent linear numeration systems, and periodic points

Contact Info

Product

Resources

About