Abstract
For each real number x ∈ (0, 1), let [a1 (x), a2 (x), ⋯, an (x), ⋯] denote its continued fraction expansion. We study the convergence exponent defined by
which reflects the growth rate of the product of two consecutive partial quotients. As a main result, the Hausdorff dimensions of the level sets of τ(x) are determined.
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The research was supported by the Scientific Research Fund of Hunan Provincial Education Department (21B0070), the Natural Science Foundation of Jiangsu Province (BK20231452), the Fundamental Research Funds for the Central Universities (30922010809) and the National Natural Science Foundation of China (11801591, 11971195, 12071171, 12171107, 12201207, 12371072).
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Fang, L., Ma, J., Song, K. et al. Multifractal analysis of convergence exponents for products of consecutive partial quotients in continued fractions. Acta Math Sci 44, 1594–1608 (2024). https://doi.org/10.1007/s10473-024-0422-6
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DOI: https://doi.org/10.1007/s10473-024-0422-6