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- Title
Gauss–Kuzmin Problem for the Difference Engel-Series Representation of Real Numbers.
- Authors
Moroz, M. P.
- Abstract
Let x = Δ g 1 x ... g n x ... E ¯ be the difference Engel-series representation of a real number x ∈ (0; 1] ( E ¯ -representation), where Δ g 1 ... g n ... E ¯ = ∑ n = 1 ∞ 1 2 + g 1 ... 2 + g 1 + ... + g n , ω n x = Δ g n + 1 x g n + 2 x ... E ¯ , is an n-fold operator of the left shift of digits in the E ¯ -representation of the number x. For a sequence of sets En(a) = {x : x ∈ (0; 1), 휔n(x) < a}, where a is a fixed parameter from (0; 1], it is proved that limn→∞λ(En(a)) = 1, where λ(∙) is a Lebesgue measure. This problem is similar to the classical Gauss–Kuzmin problem for elementary continued fractions. However, their solutions are noticeably different.
- Subjects
REAL numbers; LEBESGUE measure
- Publication
Ukrainian Mathematical Journal, 2022, Vol 74, Issue 7, p1149
- ISSN
0041-5995
- Publication type
Article
- DOI
10.1007/s11253-022-02126-x