METRIC THEORY OF PARTIAL QUOTIENTS OF N-CONTINUED FRACTIONS

Jinfeng Wang () and Yuan Zhang
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Jinfeng Wang: School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, P. R. China
Yuan Zhang: Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Benbu 233030, P. R. China

FRACTALS (fractals), 2022, vol. 30, issue 01, 1-16

Abstract: On analogy of the regular continued fractions, for any fixed positive integer N ∈ ℕ, every x ∈ [0, 1) can be expanded into an N-continued fraction, denoted by x = [a1(x),a2(x),…]N, where an(x) are called the partial quotients. In this paper, we concern the metric theory of the partial quotients. More precisely, let ϕ: ℕ → (0,∞), the Borel–Bernstein theorem and Hausdorff dimension of the set {x ∈ [0, 1): an(x) ≥ ϕ(n) for infinitely many n ∈ ℕ} are determined. This generalizes the results of regular continued fractions.

Keywords: N-Continued Fractions; Borel–Bernstein Theorem; Hausdorff Dimension (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22500220

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