Construction of $$\mu $$ μ -normal sequences

Madritsch, Manfred G.; Mance, Bill

doi:10.1007/s00605-015-0837-1

Cited by 10 publications

(12 citation statements)

References 30 publications

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“…To produce their desired CF-normal number x, they concatenated the strings X i in succession. (This technique is generalizable to many, many other systems, as demonstrated by Madritsch and Mance [11], although curiously, they seem to have been unaware of Postnikov and Pyateckii's work.) Unfortunately, the computation of the strings X i is not nearly as elegant as Champernowne's simple construction.…”

Section: Introductionmentioning

confidence: 90%

New normality constructions for continued fraction expansions

Vandehey

2016

Journal of Number Theory

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Abstract. Adler, Keane, and Smorodinsky showed that if one concatenates the finite continued fraction expansions of the sequence of rationals 1 2 , 1 3 , 2 3 , 1 4 , 2 4 , 3 4 , 1 5 , · · · into an infinite continued fraction expansion, then this new number is normal with respect to the continued fraction expansion. We show a variety of new constructions of continued fraction normal numbers, including one generated by the subsequence of rationals with prime numerators and denominators: 2 3 , 2 5 , 3 5 , 2 7 , 3 7 , 5 7 , · · · .

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Section: Introductionmentioning

confidence: 90%

New normality constructions for continued fraction expansions

Vandehey

2016

Journal of Number Theory

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“…Their continued fraction normal number is then obtained by concatenating thefinite -continued fraction expansions of these rationals [10], introduces a generalised form of normality. Neither of these works construct an explicit number that is continued fraction normal.…”

Section: Normality and Continued Fractionsmentioning

confidence: 99%

“…It took about 30 years before the constructions of Postnikov and Adler, Keane and Smorodinsky were generalized. The generalisation of Postnikov's construction, due to Madritsch and Mance [10], introduces a generalised form of normality. Neither of these works construct an explicit number that is continued fraction normal.…”

Section: Normality and Continued Fractionsmentioning

confidence: 99%

Introducing Minkowski normality

Dajani

Lepper

Robinson

2020

Journal of Number Theory

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We introduce the concept of Minkowski normality, a different type of normality for the regular continued fraction expansion. We use the ordering 1 2 ,of rationals obtained from the Kepler tree to give a concrete construction of an infinite continued fraction whose digits are distributed according to the Minkowski question mark measure. To do this we define an explicit correspondence between continued fraction expansions and binary codes to show that we can use the dyadic Champernowne number to prove normality of the constructed number. Furthermore, we provide a generalised construction based on the underlying structure of the Kepler tree, which shows that any construction that concatenates the continued fraction expansions of all rationals, ordered so that the sum of the digits of the continued fraction expansion are non-decreasing, results in a number that is Minkowski normal.

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“…In a more general setting, such symbolic shifts correspond to generalized Lüroth series. As such, although normal points x ∈ X have full measure by Birkhoff's pointwise ergodic theorem, if an explicit construction of such a point is desired, then examples can be found in [1,4,8].…”

Section: Preliminariesmentioning

confidence: 99%

Towards a sharp converse of Wall’s theorem on arithmetic progressions

Vandehey

2019

Pacific J. Math.

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Wall's theorem on arithmetic progressions says that if 0.a1a2a3 . . . is normal, then for any k, ℓ ∈ N, 0.a k a k+ℓ a k+2ℓ . . . is also normal. We examine a converse statement and show that if 0.an 1 an 2 an 3 . . . is normal for periodic increasing sequences n1 < n2 < n3 < . . . of asymptotic density arbitrarily close to 1, then 0.a1a2a3 . . . is normal. We show this is close to sharp in the sense that there are numbers 0.a1a2a3 . . . that are not normal, but for which 0.an 1 an 2 an 3 . . . is normal along a large collection of sequences whose density is bounded a little away from 1.

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Construction of $$\mu $$ μ -normal sequences

Cited by 10 publications

References 30 publications

New normality constructions for continued fraction expansions

New normality constructions for continued fraction expansions

Introducing Minkowski normality

Towards a sharp converse of Wall’s theorem on arithmetic progressions

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