Dynamical directions in numeration

Barat, Guy; Berthé, Valérie; Liardet, Pierre; Thuswaldner, Jörg Μ.

doi:10.5802/aif.2233

Cited by 60 publications

(66 citation statements)

References 254 publications

(258 reference statements)

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“…In the past few years, extensive research has been done and interesting results concerning SRS-s have been obtained, by Akiyama [60]. See also the survey papers [69,132].…”

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confidence: 99%

30 years of collaboration

Fuchs

Hajdu

2016

Period Math Hung

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Abstract. We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem), which will be presented in turn.

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“…In the past few years, extensive research has been done and interesting results concerning SRS-s have been obtained, by Akiyama [60]. See also the survey papers [69,132].…”

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confidence: 99%

30 years of collaboration

Fuchs

Hajdu

2016

Period Math Hung

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“…Let us briefly mention two other classical dynamical numeration systems (see also [7,38,43] and the references therein). Canonical number systems allow one to expand numbers in algebraic bases, see e.g., [59,56].…”

Section: Some Examples Of Numeration Dynamical Systemsmentioning

confidence: 99%

“…This transformation is usually called "successor function", "odometer " or else "adding machine". It has received considerable attention in the literature from various viewpoints; see e.g., in the context of numeration dynamics [49] and Section 5 in [7], or in the context of automata and language theory [42,13]. Note also that the representation of arbitrarily large numbers requires often the iteration of a recursive algorithmic process.…”

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confidence: 99%

“…Note also that the representation of arbitrarily large numbers requires often the iteration of a recursive algorithmic process. Numeration and dynamical systems are thus closely related such as illustrated in [7,30], which is why numeration dynamics aims at developing a dynamical viewpoint on numeration. Numeration dynamics also enters the framework of arithmetic dynamics.…”

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confidence: 99%

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Numeration and discrete dynamical systems

Berthé

2011

Computing

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This survey aims at giving both a dynamical and computer arithmetic-oriented presentation of several classical numeration systems, by focusing on the discrete dynamical systems that underly them: this provides simple algorithmic generation processes, information on the statistics of digits, on the mean behavior, and also on periodic expansions (whose study is motivated, among other things, by finite machine simulations). We consider numeration systems in a broad sense, that is, representation systems of numbers that also include continued fraction expansions. These numeration systems might be positional or not, provide unique expansions or be redundant. Special attention will be payed to β-numeration (one expands a positive real number with respect to the base β > 1), to continued fractions, and to their Lyapounov exponents. In particular, we will compare both representation systems with respect to the number of significant digits required to go from one type of expansion to the other one, through the discussion of extensions of Lochs' theorem.

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“…The number 10 is only one out of infinitely many possibilities to be a base of the radix representation of integers. This concept has far reaching generalizations, see [1], [5], [14] and the references therein. One of the successful generalizations is the concept of canonical number system polynomials with integer coefficients, which will be called in the sequel CNS polynomials.…”

Section: Introductionmentioning

confidence: 99%