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<p>This paper is concerned with the Diophantine properties of the orbits of any given point under beta-transformations as beta varies. More precisely, let <i>T</i><sub><i>β</i></sub> be the beta-transformation with parameter <i>β</i> &gt; 1 and ϕ: ℕ → [0, 1] be a non-negative function. It is shown that for any given point <i>z</i> ∈ (0, 1], for almost all or almost no parameters <i>β</i> &gt; 1, the orbit of <i>z</i> under <i>T</i><sub><i>β</i></sub> can be <i>ϕ</i>-well approximated by a given sequence {<i>x</i><sub><i>n</i></sub>}<sub><i>n</i>≥1</sub> infinitely many times according to the divergence or convergence of the series Σ<i>ϕ</i>(<i>n</i>) This can be viewed as a Duffin–Schaeffer type problem for the action of <i>β</i>-transformations compared with the action of irrational rotations. The new fundamental properties and the method presented here can also be applied to problems in beta-expansion with a fixed <i>β</i>.</p>

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Diophantine approximation of the orbits of any given point under the family of beta-transformations

  • 期刊论文

  • 发表日期：

    2025年1月20日

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  • Fan Lü,
  • Baowei Wang,
  • Jun Wu

摘要

This paper is concerned with the Diophantine properties of the orbits of any given point under beta-transformations as beta varies. More precisely, let Tβ be the beta-transformation with parameter β > 1 and ϕ: ℕ → [0, 1] be a non-negative function. It is shown that for any given point z ∈ (0, 1], for almost all or almost no parameters β > 1, the orbit of z under Tβ can be ϕ-well approximated by a given sequence {xn}n≥1 infinitely many times according to the divergence or convergence of the series Σϕ(n) This can be viewed as a Duffin–Schaeffer type problem for the action of β-transformations compared with the action of irrational rotations. The new fundamental properties and the method presented here can also be applied to problems in beta-expansion with a fixed β.