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Multiplicative Diophantine Approximation Of The Expansions Under Different Bases On A Line

Author

Listed:
  • YUAN ZHANG

    (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China)

  • MEIYING LÃœ

    (School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P. R. China)

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.

Suggested Citation

  • Yuan Zhang & Meiying Lãœ, 2021. "Multiplicative Diophantine Approximation Of The Expansions Under Different Bases On A Line," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(04), pages 1-6, June.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:04:n:s0218348x21501024
    DOI: 10.1142/S0218348X21501024
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