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Mass transference principle for limsup sets generated by rectangles

Published online by Cambridge University Press:  05 February 2015

BAO-WEI WANG ,
JUN WU  and
JIAN XU
BAO-WEI WANG
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China. e-mail: bwei_wang@hust.edu.cn; jun.wu@mail.hust.edu.cn; arielxj@hotmail.com
JUN WU
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China. e-mail: bwei_wang@hust.edu.cn; jun.wu@mail.hust.edu.cn; arielxj@hotmail.com
JIAN XU*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China. e-mail: bwei_wang@hust.edu.cn; jun.wu@mail.hust.edu.cn; arielxj@hotmail.com
*
†Corresponding author.
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Abstract

We generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set

\begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*}
we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set
\begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*}
 where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 3 , May 2015 , pp. 419 - 437
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

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