Computable absolutely Pisot normal numbers

Madritsch, Manfred G.; Scheerer, Adrian-Maria; Tichy, Robert F.

doi:10.4064/aa8661-8-2017

Cited by 9 publications

(5 citation statements)

References 24 publications

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“…For example, one can consider normal continued fractions, where the "expected" number of occurences of each partial quotient is prescribed by the Gauss-Kuzmin measure; see for example [1,49]. Other generalizations consider for example normality with respect to β-expansions [39,173], a numeration system which generalizes the b-ary expansion to non-integral bases β, or normality with respect to Cantor expansions [3,109,175], a numeration system which allows a different set of "digits" at each position. For a particularly general framework, see [172].…”

Section: Normality and Pseudorandomnessmentioning

confidence: 99%

Lacunary sequences in analysis, probability and number theory

Aistleitner¹,

Berkes²,

Tichy³

2023

Preprint

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In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudorandomness, Diophantine equations, and the subsequence principle. In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin's resonance theorem for convergence systems. More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.Contents 24 8. The subsequence principle 28 9. New results: Exact criteria for the central limit theorem for subsequences 36 Acknowledgments 49 References 50

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Section: Normality and Pseudorandomnessmentioning

confidence: 99%

Lacunary sequences in analysis, probability and number theory

Aistleitner¹,

Berkes²,

Tichy³

2023

Preprint

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“…Conversely, construction of normal sequences (as opposed to selecting normal sequences from other normal ones) has been investigated thoroughly for more than a hundred years [60,20,42,65,39,51], including explicit construction of real numbers with normal expansion for any integer base b ≥ 2 [34,55,3], and real numbers with normal expansion in non-integer bases [64,37]. Among this work, the result of most use to the present paper is the construction by Madritsch and Mance of generic sequences for any shift-invariant probability measure µ [38] -these are essentially sequences that are µ-distributed using the terminology of the present paper (see Definition 4).…”

Section: Agafonov's Theorem and Its Generalizationsmentioning

confidence: 99%

Agafonov's Theorem for finite and infinite alphabets and probability distributions different from equidistribution

Seiller¹,

Simonsen²

2020

Preprint

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An infinite sequence over a finite alphabet of symbols Σ is called normal iff the limiting frequency of every finite string w ∈ Σ * exists and equals |Σ| −|w| .A celebrated theorem by Agafonov states that a sequence α is normal iff every finitestate selector (i.e., a DFA accepting or rejecting prefixes of α) selects a normal sequence from α.Let µ : Σ * −→ [0, 1] be a probability map (for every n ≥ 0, w∈Σ n µ(w) = 1). Say that an infinite sequence α is is µ-distributed if, for every w ∈ Σ * , the limiting frequency of w in α exists and equals µ(w). Thus, α is normal if it is µ-distributed for the probability map µ(w) = |Σ| −|w| .Unlike normality, µ-distributedness is not preserved by finite-state selectors for all probability maps µ. This raises the question of how to characterize the probability maps µ for which µ-distributedness is preserved across finite-state selection, or equivalently, by selection by programs using constant space.We prove the following result: For any finite or countably infinite alphabet Σ, every finite-state selector over Σ selects a µ-distributed sequence from every µ-distributed sequence α iff µ is induced by a Bernoulli distribution on Σ, that is, for every wordThe primary -and remarkable -consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which Agafonov-type results hold. The main positive takeaway is that (the appropriate generalization of) Agafonov's Theorem holds for Bernoulli distributions (rather than just equidistributions) on both finite and countably infinite alphabets.As a further consequence, we obtain a result in the area of symbolic dynamical systems: the shift-invariant measures ν on Σ ω such that any finite-state selector preserves the property of genericity for ν are exactly the positive Bernoulli measures.

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“…Based on this construction Turing gave the first algorithm to compute an absolutely normal number [4,23]. A current research line aims to effectivize results in number theory and give algorithms to compute absolutely normal numbers that have also some other mathematical properties [6,12,18,22]. It is an open question whether there exists a fast algorithm that computes an absolutely normal number with fast speed of convergence to normality [7,17,20].…”

Section: Abstracts Of Tutorialsmentioning

confidence: 99%

2017 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM ’17 Stockholm, Sweden August 14–20, 2017

2018

Bull. symb. log

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The conferences centered on the classical subjects of mathematical and philosophical logic, as well as on many connections between these subjects and computer science. A distinctive feature of the conference was a special LC2017-CSL2017 highlight session organized on the morning of August 20th, at which each of the two conferences invited two highlight speakers to present highlights of their subject intended for the broader community represented by the two conferences. The speakers at this session were as follows: Verónica Becher (University of Buenos Aires), Normal numbers, logic and automata.

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Computable absolutely Pisot normal numbers

Abstract: We analyze the convergence order of an algorithm producing the digits of an absolutely normal number. Furthermore, we introduce a stronger concept of absolute normality by allowing Pisot numbers as bases, which leads to expansions with non-integer bases.

Cited by 9 publications

References 24 publications

Lacunary sequences in analysis, probability and number theory

Lacunary sequences in analysis, probability and number theory

Agafonov's Theorem for finite and infinite alphabets and probability distributions different from equidistribution

2017 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM ’17 Stockholm, Sweden August 14–20, 2017

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