On the construction of absolutely normal numbers

Aistleitner, Christoph; Becher, Verónica; Scheerer, Adrian-Maria; Slaman, Theodore A.

doi:10.4064/aa170213-5-8

Cited by 4 publications

(5 citation statements)

References 10 publications

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“…for all integer bases b ≥ 2; this is considered to be an "unexpectedly small" order of the discrepancy by Bugeaud in his MathSciNet review of [12]. It is not known if the exponent −1/2 of N in this estimate is optimal or not; indeed, no non-trivial lower bounds whatsoever (beyond the general lower bound (log N)/N of Schmidt) are known for this problem, but it is quite possible that whenever simultaneous normality with respect to different (multiplicatively independent) bases is considered, there must be at least one base for which the discrepancy is "large".…”

Section: Normality and Pseudorandomnessmentioning

confidence: 99%

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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%

“…The problem of the minimal order of the discrepancy of normal numbers seems to be very difficult when different bases are considered simultaneously. Aistleitner, Becher, Scheerer and Slaman [12] constructed a number x such that…”

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%

“…One possible reason for this is that only the short version (without proofs or explanation of techniques) of Agafonov's result[46] appeared in English as[1]; in contrast, the original longer paper in Russian[45] was published in a more obscure journal, and was never translated 3. In fact, one of the strategies considered by Merkle and Riemann, which consists in computing the language {ww R | w ∈ Σ * } where w R is the reverse of w, can be computed by an arguably less expressive model of computation, namely aDFA(2) -two-way automata with two heads[28].…”

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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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“…where gcd(b, c) = 1, and s ∈ R, see [15], [3], et alii. More complex constructions and algorithms for generating normal numbers are developed in [2], [6], et alii. However, there are no known normal numbers in closed forms such as √ 2, e, π, log 2, γ, ..., et cetera in any base b ≥ 2.…”

Section: Introduction To Normal Numbersmentioning

confidence: 99%