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    A divergent Khintchine theorem in the real, complex, and p-adic fields


    Bernik, Vasili and Budarina, Natalia and Dickinson, Detta (2008) A divergent Khintchine theorem in the real, complex, and p-adic fields. Lithuanian Mathematical Journal, 48 (2). pp. 158-173. ISSN 0363-1672

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    Abstract

    In this paper, we show that if the sum ∑ r=1 ∞ Ψ(r) diverges, then the set of points (x, z, w) ∈ ℝ × ℂ × ℚp satisfying the inequalities |P(x)|<H−v1Ψλ1(H),|P(z)|<H−v2Ψλ2(H), and |P(w)|p<H−v3Ψλ3(H) for infinitely many integer polynomials P has full measure. With a special choice of parameters v i and λ i , i = 1, 2, 3, we can obtain all the theorems in the metric theory of transcendental numbers which were known in the real, complex, or p-adic fields separately.

    Item Type: Article
    Additional Information: Cite as: Bernik, V., Budarina, N. & Dickinson, D. Lith Math J (2008) 48: 158. https://doi.org/10.1007/s10986-008-9005-9
    Keywords: Diophantine approximation; Khintchine-type theorems metric; theory of transcendental numbers;
    Academic Unit: Faculty of Science and Engineering > Mathematics and Statistics
    Item ID: 10111
    Identification Number: https://doi.org/10.1007/s10986-008-9005-9
    Depositing User: Dr. Detta Dickinson
    Date Deposited: 16 Oct 2018 17:04
    Journal or Publication Title: Lithuanian Mathematical Journal
    Publisher: Springer
    Refereed: Yes
    URI:

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