MULTIPLICATIVE DIOPHANTINE APPROXIMATION OF THE EXPANSIONS UNDER DIFFERENT BASES ON A LINE

Yuan Zhang () and LÜ Meiying
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Yuan Zhang: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China
LÜ Meiying: School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 04, 1-6

Abstract: In this paper, we discuss a problem of multiplicative Diophantine approximation of the expansions under different bases restricted on a line. More specifically, let φ : ℕ → [0,∞) be a positive function. A dichotomy law of the Hausdorff measure for the following set: M(φ) = {x ∈ [0, 1] : ∥2nx∥⋅∥3nx∥ < φ(n) for i.m. n ∈ ℕ}, is obtained, which depends the convergence or divergence of certain series.

Keywords: Multiplicatively Diophantine Approximation; Hausdorff Measure (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21501024

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