Mass transference principle for limsup sets generated by rectangles

Abstract

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1be a sequence of points in the unit cube [0, 1]dwithd⩾ 1 and {rn}n⩾1a 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*}wherea= (a1, . . .,ad) with 1 ⩽a1⩽a2⩽ . . . ⩽adandBa(x, r) denotes a rectangle with centerxand side-length (ra1,ra2,. . .,rad). Whena1=a2= . . . =ad, the result is included in the setting considered by Beresnevich and Velani.