Abstract
We show that limsup sets generated by a sequence of open sets in compact Ahlfors s-regular \((0<s<\infty )\) space \((X,{\mathscr{B}},\mu ,\rho )\) belong to the classes of sets with large intersections with index λ, denoted by \(\mathcal {G}^{\lambda }(X)\), under some conditions. In particular, this provides a lower bound on Hausdorff dimension of such sets. These results are applied to obtain that limsup random fractals with indices γ2 and δ belong to \(\mathcal {G}^{s-\delta -\gamma _{2}}(X)\) almost surely, and random covering sets with exponentially mixing property belong to \(\mathcal {G}^{s_{0}}(X)\) almost surely, where s0 equals to the corresponding Hausdorff dimension of covering sets almost surely. We also investigate the large intersection property of limsup sets generated by rectangles in metric space.
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References
Aubry, J. M., Jaffard, S.: Random wavelet series. Comm. Math. Phys. 227(3), 483–514 (2002)
Beresnevich, V., Velani, S.: A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3), 971–992 (2006)
Bugeaud, Y.: Intersective sets and Diophantine approximation. Michigan Math. J. 52(3), 667–682 (2004)
Ding, N.: Large intersection properties for limsup sets generated by rectangles in compact metric spaces. Fractals 29(6), 2150137, 13 (2021)
Durand, A.: Sets with large intersection and ubiquity. Math. Proc Cambridge Philos Soc. 144(1), 119–144 (2008)
Durand, A.: Large intersection properties in Diophantine approximation and dynamical systems. J. Lond. Math. Soc. (2) 79(2), 377–398 (2009)
Durand, A.: Singularity sets of lévy processes. Probab. Theory Related Fields 143(3-4), 517–544 (2009)
Durand, A.: On Randomly Placed Arcs on the Circle. Recent Developments in Fractals and Related Fields, 343–351, Appl. Numer. Harmon. Anal. Birkhäuser, Boston (2010)
Falconer, K.: Sets with large intersection Properties. J. London Math. Soc. (2) 49(2), 267–280 (1994)
Färm, D., Persson, T.: Large intersection classes on fractals. Nonlinearity 24(4), 1291–1309 (2011)
Fraser, J.: Assouad Dimension and Fractal Geometry (Cambridge Tracts in Mathematics). Cambridge University Press, Cambridge (2020)
Heinonen, J.: Lectures on Analysis on Metric Spaces, Universitext. Springer, New York (2001)
Hill, R., Velani, S.: The ergodic theory of shrinking targets. Invent Math. 119(1), 175–198 (1995)
Hu, Z. -N., Cheng, W.-C., Li, B.: On the hitting probabilities of limsup random fractals. Fractals 30(3), 2250055 (2022)
Hu, Z. -N., Li, B.: Random covering sets in metric space with exponentially mixing property. Statist. Probab. Lett. 168, 108922 7 (2021)
Hu, Z. -N., Li, B., Xiao, Y. -M.: On the intersection of dynamical covering sets with fractals. Math Z 301(1), 485–513 (2022)
Käenmäki, A., Rajala, T., Suomala, V.: Existence of doubling measures via generalised nested cubes. Proc. Amer. Math. Soc. 140(9), 3275–3281 (2012)
Khoshnevisan, D., Peres, Y., Xiao, Y.-M.: Limsup random fractals. Electron. J. Probab. 5(5), 24 (2000)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)
McShane, E. J.: Jensen’s, inequality. Bull. Amer. Math Soc. 43(8), 521–527 (1937)
Negreira, F., Sequeira, E.: Sets with large intersection properties in metric spaces. J. Math. Anal. Appl. 511(1), 126064 25 (2022)
Persson, T.: Inhomogeneous potentials Hausdorff dimension and shrinking targets. Ann. H. Lebesgue 2, 1–37 (2019)
Persson, T.: A mass transference principle and sets with large intersections. Real Anal. Exchange 47(1), 191–205 (2022)
Wang, B.-W., Wu, J., Xu, J.: Mass transference principle for limsup sets generated by rectangles. Math. Proc. Cambridge Philos. Soc. 158(3), 419–437 (2015)
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We thank the referee for the helpful comments. The work was supported by NSFC 12271176, 11671151 and 11871208.
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Hu, Zn., Li, B. & Yang, L. Large Intersection Property for Limsup Sets in Metric Space. Potential Anal 60, 341–363 (2024). https://doi.org/10.1007/s11118-022-10052-7
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DOI: https://doi.org/10.1007/s11118-022-10052-7