The shrinking target problem in the dynamical system of continued fractions
The shrinking target problem in the dynamical system of continued fractions
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DOI：
10.1112/plms/pdt017
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发表时间：
2014-01
影响因子：
1.8
通讯作者：
Bing Li;Bao-Wei Wang;Jun Wu;Jian Xu
中科院分区：
文献类型：
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作者：
Bing Li;Bao-Wei Wang;Jun Wu;Jian Xu
设（[0，1），T）为连分式动力系统.设{zn}n <$1是[0，1]中的真实的序列，且<$：<$× [0，1）→ <$+是正函数.一个点x∈[0，1）被称为可由{zn}n <$1逼近，如果|Tnx-zn| <n（n，x）对无穷多个n ∈ n成立。本文研究了可逼近点集的Hausdorff维数。当<$（n，x）=<$（n）独立于x且当<$（n，x）=e−（f（x）+<$+f（Tn−1x））且f为正连续函数时，维数完全确定。为了证明这些结果，研究了[0，1）中的球与由连分式中的部分幂次所定义的圆柱之间的关系.证明了球可以被同阶、长度相当的圆柱充分填充，从而给出了在[0，1）中求球内点的显式连分式表示.
Let ([0, 1), T) be the dynamical system of continued fractions. Let {zn}n⩾1 be a sequence of real numbers in [0, 1] and ψ: ℕ × [0, 1) → ℝ+ be a positive function. A point x∈[0, 1) is said to be ψ‐approximable by {zn}n ⩾ 1 if |Tnx − zn| < ψ(n, x) holds for infinitely many n ∈ ℕ. In this paper, the Hausdorff dimension of the set of ψ‐approximable points is studied. The dimensions are completely determined when ψ(n, x) = ψ(n) independent on x and when ψ(n,x)=e−(f(x)+⋯+f(Tn−1x)) with f a positive continuous function. For the proof of these results, a relationship between a ball in [0, 1) and the cylinders defined by the partial quotients in continued fractions is investigated. It is shown that a ball can be sufficiently packed by cylinders of the same order and of comparable length, which gives us explicit continued fraction representations in locating the points in a ball in [0, 1).