Normal Numbers and Computer Science

Becher, Verónica; Carton, Olivier

doi:10.1007/978-3-319-69152-7_7

Cited by 14 publications

(15 citation statements)

References 24 publications

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“….. The concept of normality of numbers was introduced by Borel [7] in 1909 and there are several equivalent formulations (see [4,8]). The most convenient for our purposes was given by Pillai [15]: A real number x is simply normal to a given base b if every possible digit in {0, 1, .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…We hope that Theorem 4 will help to find a construction of a normal sequence in T P for some general finite set P of primes. A construction of one explicit normal sequence in T P for b = 2 and the special case P = {2} appears in [4] and [5]. This construction can be generalized to any integer base b and any singleton P .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Normal numbers with digit dependencies

Aistleitner

Becher

Carton

2018

Trans. Amer. Math. Soc.

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We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that, still, almost all real numbers are normal. Our theorem states that almost all real numbers are normal when at least slightly more than log log n consecutive digits with indices starting at position n are independent. As the main application, we consider the Toeplitz set T P , which is the set of all sequences a 1 a 2 . . . of symbols from {0, . . . , b − 1} such that an is equal to apn, for every p in P and n = 1, 2, . . .. Here b is an integer base and P is a finite set of prime numbers. We show that almost every real number whose base b expansion is in T P is normal to base b. In the case when P is the singleton set {2} we prove that more is true: almost every real number whose base b expansion is in T P is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set T P and we characterize the normal sequences whose Toeplitz transform is normal as well.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Normal numbers with digit dependencies

Aistleitner

Becher

Carton

2018

Trans. Amer. Math. Soc.

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“…With this notation we can state the formulation of normality that is most convenient to solve our problem. A thorough presentation of normality can be read from the monographs [2,4].…”

Section: Tools and Lemmasmentioning

confidence: 99%

From Randomness in Two Symbols to Randomness in Three Symbols

Zylber

2021

Uniform Distribution Theory

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In 1909 Borel defined normality as a notion of randomness of the digits of the representation of a real number over certain base (fractional expansion). If we think of the representation of a number over a base as an infinite sequence of symbols from a finite alphabet A, we can define normality directly for words of symbols of A: A word x is normal to the alphabet A if every finite block of symbols from A appears with the same asymptotic frequency in x as every other block of the same length. Many examples of normal words have been found since its definition, being Champernowne in 1933 the first to show an explicit and simple instance. Moreover, it has been characterized how we can select subsequences of a normal word x preserving its normality, always leaving the alphabet A fixed. In this work we consider the dual problem which consists of inserting symbols in infinitely many positions of a given word, in such a way that normality is preserved. Specifically, given a symbol b that is not present in the original alphabet A and given a word x that is normal to the alphabet A we solve how to insert the symbol b in infinitely many positions of the word x such that the resulting word is normal to the expanded alphabet A ∪{b}.

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“…An occurrence of v is called aligned if its starting position i (as above) is such that i − 1 is a multiple of the length of v. An alternative definition of normality can be given by counting aligned occurrences, and it is well-known that they are equivalent (see for example [2]). We refer the reader to [5,Chap.4] for a complete introduction to normality.…”

Section: Basic Definitions 21 Normalitymentioning

confidence: 99%

Preservation of normality by unambiguous transducers

Carton¹

2020

Preprint

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We consider finite state non-deterministic but unambiguous transducers with infinite inputs and infinite outputs, and we consider the property of Borel normality of sequences of symbols. When these transducers are strongly connected, and when the input is a Borel normal sequence, the output is a sequence in which every block has a frequency given by a weighted automaton over the rationals. We provide an algorithm that decides in cubic time whether a unambiguous transducer preserves normality.

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Normal Numbers and Computer Science

Cited by 14 publications

References 24 publications

Normal numbers with digit dependencies

Normal numbers with digit dependencies

From Randomness in Two Symbols to Randomness in Three Symbols

Preservation of normality by unambiguous transducers

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