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Mass transference principle for limsup sets generated by rectangles

Published online by Cambridge University Press:  05 February 2015

School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China. e-mail:;;
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China. e-mail:;;
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China. e-mail:;;
Corresponding author.


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 ⩽ a1a2 ⩽ . . . ⩽ 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.

Research Article
Copyright © Cambridge Philosophical Society 2015 

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