A function whose graph has positive doubling measure
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- by Tuomo Ojala and Tapio Rajala PDF
- Proc. Amer. Math. Soc. 144 (2016), 733-738 Request permission
Abstract:
We show that a doubling measure on the plane can give positive measure to the graph of a continuous function. This answers a question by Wang, Wen and Wen posed in a 2013 paper. Moreover, we show that the doubling constant of the measure can be chosen to be arbitrarily close to the doubling constant of the Lebesgue measure.References
- 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
- Fengji Peng and Shengyou Wen, On Sierpiński carpets and doubling measures, Nonlinearity 27 (2014), no. 6, 1287–1298. MR 3207934, DOI 10.1088/0951-7715/27/6/1287
- 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
Additional Information
- Tuomo Ojala
- Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland
- MR Author ID: 908269
- Email: tuomo.j.ojala@jyu.fi
- Tapio Rajala
- Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland
- MR Author ID: 838027
- Email: tapio.m.rajala@jyu.fi
- Received by editor(s): January 28, 2015
- Received by editor(s) in revised form: February 3, 2015
- Published electronically: June 30, 2015
- Communicated by: Jeremy Tyson
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 733-738
- MSC (2010): Primary 28A12; Secondary 30L10
- DOI: https://doi.org/10.1090/proc12748
- MathSciNet review: 3430849