A divergent Khintchine theorem in the real,complex, and <Emphasis Type="Italic">p</Emphasis>-adic fields期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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A divergent Khintchine theorem in the real,complex, and <Emphasis Type="Italic">p</Emphasis>-adic fields

Authors:	V Bernik N Budarina D Dickinson

Institution:	(1) Institute of Mathematics, National Academy of Sciences of Belarus, Surganova 11, 220072 Minsk, Belarus;(2) Department of Mathematics, National University of Ireland, Maynooth, Co. Kildare, Republic of Ireland

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 × ℚ_p satisfying the inequalities $\left\| {P(x)} \right\| < H^{ - v_1 } \Psi ^{\lambda _1 } (H),\left\| {P(z)} \right\| < H^{ - v_2 } \Psi ^{\lambda _2 } (H)$ , and $\left\| {P(w)} \right\|p < H^{ - v_3 } \Psi ^{\lambda _3 } (H)$ 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.

Keywords:	Diophantine approximation Khintchine-type theorems metric theory of transcendental numbers
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