Lüroth series and their ergodic properties

Jager, H.; Vroedt, C. de

doi:10.1016/1385-7258(69)90023-7

Cited by 43 publications

(26 citation statements)

References 2 publications

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“…The first class include the representations of numbers by s-adic fractions, Q-representations [1], etc. Among the most well-known representations with infinite alphabets, one can mention the representations of real numbers by the Ostrogradskii series (of the first and second kinds) [2], Lüroth series [3], etc. This class of representations also includes the procedure of representation of real numbers by elementary continued fractions whose elements are natural numbers.…”

Section: Introductionmentioning

confidence: 99%

A 2-continued fraction representation of real numbers and its geometry

Dmytrenko¹,

Kyurchev²,

Prats’ovytyĭ

2009

Ukr Math J

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We study the geometry of representations of numbers by continued fractions whose elements belong to the set A 2. It is shown that, for α α 1 2 1 2 ≤ / , every point of a certain segment admits an A 2 -continued fraction representation.Moreover, for α α 1 2 1 2 = / , this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose A 2 -continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its A 2 -continued fraction representation form a homogeneous Markov chain are also investigated.

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Section: Introductionmentioning

confidence: 99%

A 2-continued fraction representation of real numbers and its geometry

Dmytrenko¹,

Kyurchev²,

Prats’ovytyĭ

2009

Ukr Math J

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“…By using standard measure theory as in [4], it follows that the same equation holds for any Borel set E in I. Now let E be a Borel set in I such that T −1 E = E, where T is a measurable, nonsingular transformation.…”

Section: Proof Define a Linear Mapmentioning

confidence: 99%

On the application of ergodic theory to alternating Engel series

Ganatsiou

2001

International Journal of Mathematics and Mathematical Sciences

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“…The metric and ergodic properties of the sequence {d n (x)} n 1 and the Lüroth map T defined by (1.1) have been extensively studied in [4] (see also [7], [8], [9], [12], [16]). The behavior of approximating real numbers by the Lüroth expansion was thoroughly investigated in [2], [3], where the authors studied the distribution of the approximation coefficients θ n = θ n (x) = Q n (x)x − P n (x) for n 1.…”

Section: Introductionmentioning

confidence: 99%