On thin carpets for doubling measures
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- by Changhao Chen and Shengyou Wen PDF
- Proc. Amer. Math. Soc. 147 (2019), 3439-3449 Request permission
Abstract:
We study thin sets for doubling or isotropic doubling measures in $\mathbb {R}^{d}$. In our results we prove that the self-affine sets satisfying the open set condition with holes are thin for isotropic doubling measures and among them Barański carpets are thin for doubling measures.References
- Krzysztof Barański, Hausdorff dimension of the limit sets of some planar geometric constructions, Adv. Math. 210 (2007), no. 1, 215–245. MR 2298824, DOI 10.1016/j.aim.2006.06.005
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- John Garnett, Rowan Killip, and Raanan Schul, A doubling measure on $\Bbb R^d$ can charge a rectifiable curve, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1673–1679. MR 2587452, DOI 10.1090/S0002-9939-10-10234-2
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- Dan Han, Lisha Wang, and Shengyou Wen, Thickness and thinness of uniform Cantor sets for doubling measures, Nonlinearity 22 (2009), no. 3, 545–551. MR 2480101, DOI 10.1088/0951-7715/22/3/002
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Antti Käenmäki, Tapio Rajala, and Ville Suomala, Existence of doubling measures via generalised nested cubes, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3275–3281. MR 2917099, DOI 10.1090/S0002-9939-2012-11161-X
- Leonid Kovalev, Diego Maldonado, and Jang-Mei Wu, Doubling measures, monotonicity, and quasiconformality, Math. Z. 257 (2007), no. 3, 525–545. MR 2328811, DOI 10.1007/s00209-007-0132-5
- Jouni Luukkainen and Eero Saksman, Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc. 126 (1998), no. 2, 531–534. MR 1443161, DOI 10.1090/S0002-9939-98-04201-4
- Tuomo Ojala, On $(\alpha _n)$-regular sets, Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 2, 655–673. MR 3237043, DOI 10.5186/aasfm.2014.3926
- Tuomo Ojala and Tapio Rajala, A function whose graph has positive doubling measure, Proc. Amer. Math. Soc. 144 (2016), no. 2, 733–738. MR 3430849, DOI 10.1090/proc12748
- Tuomo Ojala, Tapio Rajala, and Ville Suomala, Thin and fat sets for doubling measures in metric spaces, Studia Math. 208 (2012), no. 3, 195–211. MR 2911493, DOI 10.4064/sm208-3-1
- A. L. Vol′berg and S. V. Konyagin, On measures with the doubling condition, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 666–675 (Russian); English transl., Math. USSR-Izv. 30 (1988), no. 3, 629–638. MR 903629
- Wen Wang, Shengyou Wen, and Zhi-Ying Wen, Fat and thin sets for doubling measures in Euclidean space, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 2, 535–546. MR 3113093, DOI 10.5186/aasfm.2013.3827
- Wen Wang, Shengyou Wen, and Zhixiong Wen, Note on atomic doubling measures, quasisymmetrically thin sets and thick sets, J. Math. Anal. Appl. 385 (2012), no. 2, 1027–1032. MR 2834908, DOI 10.1016/j.jmaa.2011.07.026
- Jang-Mei Wu, Null sets for doubling and dyadic doubling measures, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 1, 77–91. MR 1207896
- Jang-Mei Wu, Hausdorff dimension and doubling measures on metric spaces, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1453–1459. MR 1443418, DOI 10.1090/S0002-9939-98-04317-2
Additional Information
- Changhao Chen
- Affiliation: Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu, Finland
- MR Author ID: 941656
- Email: changhao.chen@unsw.edu.au
- Shengyou Wen
- Affiliation: Department of Mathematics and Key Laboratory of Applied Mathematics of Hubei University, Wuhan 430062, People’s Republic of China
- MR Author ID: 675487
- Email: sywen_65@163.com
- Received by editor(s): November 2, 2017
- Received by editor(s) in revised form: June 6, 2018, and November 14, 2018
- Published electronically: March 21, 2019
- Additional Notes: This research was supported by the Vilho, Yrjö, and Kalle Väisälä Foundation and by the NSFC (grants No. 11871200, 11271114, and 11671189).
- Communicated by: Jeremy Tyson
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3439-3449
- MSC (2010): Primary 28A80; Secondary 28A75
- DOI: https://doi.org/10.1090/proc/14493
- MathSciNet review: 3981122