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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
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.

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 ⩽ 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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1]Beresnevich, V. and Velani, S.A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164 (2006), no. 3, 971992.Google Scholar
[2]Beresnevich, V., Dickinson, D. and Velani, S.Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179 (2006), no. 846, x+91 pp.Google Scholar
[3]Bernik, V. I. and Dodson, M. M.Metric Diophantine approximation on Manifolds Cambridge Tracts in Mathematics 137 (Cambridge University Press, Cambridge, 1999), xii+172 pp.CrossRefGoogle Scholar
[4]Bovey, J. D. and Dodson, M. M.The Hausdorff dimension of systems of linear forms. Acta Arith. 45 (1986), no. 4, 337358.CrossRefGoogle Scholar
[5]Dickinson, D. and Velani, S.Hausdorff measure and linear forms. J. Reine Angew. Math. 490 (1997), 136.Google Scholar
[6]Dodson, M. M.Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation. J. Reine Angew. Math. 432 (1992), 6976.Google Scholar
[7]Dodson, M. M.Geometric and probabilistic ideas in the metric theory of Diophantine approximations. Uspekhi Mat. Nauk 48 (1993), no.5(293), 77106.Google Scholar
[8]Dodson, M. M., Rynne, B. P. and Vickers, J. A. G.Diophantine approximation and a lower bound for Hausdorff dimension. Mathematika 37 (1990), no. 1, 5973.Google Scholar
[9]Falconer, K. J.Fractal Geometry, Mathematical Foundations and Application (John Wiley & Sons, Ltd., Chichester, 1990).Google Scholar
[10]Groshev, A. V.Une théorème sur les systèmes des formes linéaires. Dokl. Akad. Nauk SSSR 9 (1938), 151152.Google Scholar
[11]Harman, G.Metric Number Theory. London Math. Series Monogr. 18 (Clarendon Press, Oxford 1998).Google Scholar
[12]Hill, R. and Velani, S.The ergodic theory of shrinking targets. Invent. Math. 119 (1995), no. 1, 175198.CrossRefGoogle Scholar
[13]Hill, R. and Velani, S.The shrinking target problem for matrix transformations of tori. J. London Math. Soc. (2) 60 (1999), no. 2, 381398.CrossRefGoogle Scholar
[14]Jarník, I.Uber die simultanen diophantischen Approximationen. Math. Z. 33 (1931) 503543.Google Scholar
[15]Pollington, A. and Vaughan, R.The k-dimensional Duffin and Schaeffer conjecture. Mathematika 37 (1990) 190200.Google Scholar
[16]Rynne, B. P.Hausdorff dimension and generalized simultaneous Diophantine approximation. Bull. London Math. Soc. 30 (1998), 365376.Google Scholar
[17]Schmidt, W. M.Diophantine approximation. Lecture Notes in Math. 785 (1980).Google Scholar
[18]Sprindzuk, V.Metric theory of Diophantine Approximations (translated by R. A. Silverman). (V. H. Winston & Sons, Washington D.C., 1979).Google Scholar