Computable absolutely normal numbers and discrepancies

Scheerer, Adrian-Maria

doi:10.1090/mcom/3189

Cited by 6 publications

(6 citation statements)

References 18 publications

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“…As reported in [11], prior to the present work the construction of an absolutely normal number with the smallest discrepancy bound was due to Levin [8]. Given a countable set L of reals greater than 1, Levin constructs a real number x such that for every θ in L,…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the construction of absolutely normal numbers

et al. 2017

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We give a construction of an absolutely normal real number x such that for every integer b greater than or equal to 2, the discrepancy of the first N terms of the sequence (b n x mod 1) n≥0 is of asymptotic order O(N −1/2 ). This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order was not known before.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the construction of absolutely normal numbers

et al. 2017

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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

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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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“…This construction was made computable by Becher and Figueira [4] who gave a recursive formulation of Sierpinski's construction. Other algorithms for constructing absolutely normal numbers are due to Turing [36] (see also Becher,Figueira and Picchi [5]), Schmidt [32] (see also Scheerer [28]) and Levin [20] (see also Alvarez and Becher [1]).…”

Section: Absolute Normality and Order Of Convergencementioning

confidence: 99%