Mass transference principle for limsup sets generated by rectangles.(English)Zbl 1371.11119
Summary: We generalise the mass transference principle established byV. Beresnevich andS. Velani [Ann. Math. (2) 164, No. 3, 971–992 (2006;Zbl 1148.11033)] to limsup sets generated by rectangles. More precisely, let \(\{x_n\}_{n\geq1}\) be a sequence of points in the unit cube \([0, 1]^d\) with \(d\geq 1\) and \(\{r_n\}_{n\geq1}\) a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \[\big\{x\in [0,1]^d: x\in B(x_n,r_n), \text{ for infinitely many } n\in \mathbb{N}\big\},\] we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \[\big\{x\in [0,1]^d: x\in B^{\mathbf{a}}(x_n,r_n), \text{ for infinitely many }n\in \mathbb{N}\big\},\] where \(\mathbf{a}=(a_1,\dots, a_d)\) with \(1\leq a_1\leq a_2\leq\dots\leq a_d\) and \(B^{\mathbf{a}}(x, r)\) denotes a rectangle with center \(x\) and side-length \((r^{a_1}, r^{a_2},\dots,r^{a_d})\). When \(a_1=a_2=\dots=a_d\), the result is included in the setting considered by Beresnevich and Velani [loc. cit.].
MSC:
| 11J83 | Metric theory |
| 28A78 | Hausdorff and packing measures |
| 11J13 | Simultaneous homogeneous approximation, linear forms |
| 11K60 | Diophantine approximation in probabilistic number theory |
| 11J04 | Homogeneous approximation to one number |
Citations:
Zbl 1148.11033Cite
Full Text:DOI
References:
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