The sets of Dirichlet non-improvable numbers vs well-approximable numbers

2018/06/02 by Ayreena Bakhtawar, Philip Bos, Philip J. Bos +4
Mathematics · #Combinatorics #Dirichlet distribution #Dynamical Systems (math.DS) #Econometrics #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical economics #Mathematics #Nonlinear system #Number Theory (math.NT) #Physics #advanced mathematical theories #math.DS #math.NT

paper · pdf · doi:10.48550/arxiv.1806.00618

17 pages, comments welcome, to appear in Ergodic Theory and Dynamical System

openalex publication_date 2018/06/02 · arxiv created 2019/05/17 · arxiv updated 2019/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ψ:[1,∞ )→ ℝ₊ be a non-decreasing function, a_n(x) the n'th partial quotient of x and q_n(x) the denominator of the n'th convergent. The set of Ψ-Dirichlet non-improvable numbers G(Ψ):=\x∈ \lbrack 0,1):a_n(x)a_n+1(x) gt; Ψ(q_n(x) ) for infinitely many n∈ ℕ\, is related with the classical set of 1/q²Ψ(q)-approximable numbers K(Ψ) in the sense that K(3Ψ)⊂ G(Ψ). Both of these sets enjoy the same s-dimensional Hausdorff measure criterion for s∈ (0,1). We prove that the set G(Ψ)∖ K(3Ψ) is uncountable by proving that its Hausdorff dimension is the same as that for the sets K(Ψ) and G(Ψ). This gives an affirmative answer to a question raised by Hussain-Kleinbock-Wadleigh-Wang (2018).

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