当前位置: X-MOL 学术Isr. J. Math. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Metric properties of the product of consecutive partial quotients in continued fractions
Israel Journal of Mathematics ( IF 1 ) Pub Date : 2020-07-01 , DOI: 10.1007/s11856-020-2049-1
Lingling Huang , Jun Wu , Jian Xu

In the one-dimensional Diophantine approximation, by using the continued fractions, Khintchine’s theorem and Jarnik’s theorem are concerned with the growth of the large partial quotients, while the improvability of Dirichlet’s theorem is concerned with the growth of the product of consecutive partial quotients. This paper aims to establish a complete characterization on the metric properties of the product of the partial quotients, including the Lebesgue measure-theoretic result and the Hausdorff dimensional result. More precisely, for any x ∈ [0, 1), let x =[a1, a2, …] beits continued fraction expansion. The size of the following set, in the sense of Lebesgue measure and Hausdorff dimension, Em(ϕ):= {x ∈ [0, 1): an (x) ⋯ an+m−1 (x) ≥ ϕ(n) for infinitely many n ∈ ℕ}, are given completely, where m ≥ 1 is an integer and ϕ: ℕ → ℝ+ is a positive function.

中文翻译:

连分数中连续偏商的乘积的度量属性

在一维丢番图近似中,通过使用连分数,Khintchine 定理和 Jarnik 定理涉及大偏商的增长,而 Dirichlet 定理的可改进性涉及连续偏商乘积的增长。本文旨在建立对偏商乘积的度量性质的完整表征,包括勒贝格测度理论结果和豪斯多夫维数结果。更准确地说,对于任何 x ∈ [0, 1),让 x =[a1, a2, …] 成为连续分数展开式。以下集合的大小,在 Lebesgue 测度和 Hausdorff 维的意义上,Em(ϕ):= {x ∈ [0, 1): an (x) ⋯ an+m−1 (x) ≥ ϕ(n)对于无穷多个 n ∈ ℕ},完全给定,其中 m ≥ 1 是一个整数和 ϕ:
更新日期:2020-07-01
down
wechat
bug