A new Hausdorff content bound for limsup sets.(English)Zbl 07839369
Summary: We give a new Hausdorff content bound for limsup sets, which is related to Falconer’s sets of large intersection. Falconer’s sets of large intersection satisfy a content bound for all balls in a space. In comparison, our main theorem only assumes a scale-invariant bound for the balls forming the limit superior set in question.
We give four applications of these ideas and our main theorem: a new proof and generalization of the mass transference principle related to Diophantine approximations, a related result on random limsup sets, a new proof of Federer’s characterization of sets of finite perimeter and a statement concerning generic paths and the measure theoretic boundary. The new general mass transference principle infers a content bound for one collection of balls from the content bound of another collection of sets. The benefit of our approach is greatly simplified arguments as well as new tools to estimate Hausdorff content.
The new methods allow for us to dispense with many of the assumptions in prior work. Specifically, our general Mass Transference Principle, and bounds on random limsup sets, do not assume Ahlfors regularity. Further, they apply to any complete metric space. This generality is made possible by the fact that our general Hausdorff content estimate applies to limsup sets in any complete metric space.
We give four applications of these ideas and our main theorem: a new proof and generalization of the mass transference principle related to Diophantine approximations, a related result on random limsup sets, a new proof of Federer’s characterization of sets of finite perimeter and a statement concerning generic paths and the measure theoretic boundary. The new general mass transference principle infers a content bound for one collection of balls from the content bound of another collection of sets. The benefit of our approach is greatly simplified arguments as well as new tools to estimate Hausdorff content.
The new methods allow for us to dispense with many of the assumptions in prior work. Specifically, our general Mass Transference Principle, and bounds on random limsup sets, do not assume Ahlfors regularity. Further, they apply to any complete metric space. This generality is made possible by the fact that our general Hausdorff content estimate applies to limsup sets in any complete metric space.
MSC:
| 28A78 | Hausdorff and packing measures |
| 28A75 | Length, area, volume, other geometric measure theory |
| 30L99 | Analysis on metric spaces |
| 28A80 | Fractals |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
| 26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |
| 31E05 | Potential theory on fractals and metric spaces |
| 54E35 | Metric spaces, metrizability |
Cite
References:
| [1] | Allen, D.; Baker, S., A general mass transference principle, Sel. Math. New Ser., 25, 3, Article 39 pp., 2019 ·Zbl 1450.11079 |
| [2] | Allen, D.; Beresnevich, V., A mass transference principle for systems of linear forms and its applications, Compos. Math., 154, 5, 1014-1047, 2018 ·Zbl 1404.11097 |
| [3] | Allen, D.; Troscheit, S., The mass transference principle: ten years on, (Horizons of Fractal Geometry and Complex Dimensions. Horizons of Fractal Geometry and Complex Dimensions, Contemp. Math., vol. 731, 2019, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-33, ©2019 ·Zbl 1423.11124 |
| [4] | Ambrosio, L., Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math., 159, 1, 51-67, 2001 ·Zbl 1002.28004 |
| [5] | Ambrosio, L., Fine properties of sets of finite perimeter in doubling metric measure spaces, Set-Valued Anal., 10, 111-128, 2002 ·Zbl 1037.28002 |
| [6] | Ambrosio, L., On some recent developments of the theory of sets of finite perimeter, Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., 14, 179-187, 2003, (2004) ·Zbl 1225.49039 |
| [7] | Beresnevich, V.; Velani, S., A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. Math. (2), 164, 3, 971-992, 2006 ·Zbl 1148.11033 |
| [8] | Cheeger, J.; Kleiner, B., Differentiating maps into \(L^1\), and the geometry of BV functions, Ann. Math. (2), 171, 2, 1347-1385, 2010 ·Zbl 1194.22009 |
| [9] | Durand, A., Sets with large intersection and ubiquity, Math. Proc. Camb. Philos. Soc., 144, 1, 119-144, 2008 ·Zbl 1239.11076 |
| [10] | Ekström, F., Dimension of random limsup sets, Am. Math. Mon., 126, 9, 816-824, 2019 ·Zbl 1429.28012 |
| [11] | Ekström, F.; Järvenpää, E.; Järvenpää, M., Hausdorff dimension of limsup sets of rectangles in the Heisenberg group, Math. Scand., 126, 2, 229-255, 2020 ·Zbl 1456.60038 |
| [12] | Ekström, F.; Järvenpää, E.; Järvenpää, M.; Suomala, V., Hausdorff dimension of limsup sets of random rectangles in products of regular spaces, Proc. Am. Math. Soc., 146, 6, 2509-2521, 2018 ·Zbl 1386.60043 |
| [13] | Ekström, F.; Persson, T., Hausdorff dimension of random limsup sets, J. Lond. Math. Soc. (2), 98, 3, 661-686, 2018 ·Zbl 1408.28012 |
| [14] | Evans, L. C.; Gariepy, R. F., Measure Theory and Fine Properties of Functions, Textbooks in Mathematics, 2015, CRC Press: CRC Press Boca Raton, FL ·Zbl 1310.28001 |
| [15] | Falconer, K. J., Classes of sets with large intersection, Mathematika, 32, 2, 191-205, 1985, (1986) ·Zbl 0606.28003 |
| [16] | Federer, H., Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, vol. 153, 1969, Springer-Verlag: Springer-Verlag New York Inc., New York ·Zbl 0176.00801 |
| [17] | Feng, D.-J.; Järvenpää, E.; Järvenpää, M.; Suomala, V., Dimensions of random covering sets in Riemann manifolds, Ann. Probab., 46, 3, 1542-1596, 2018 ·Zbl 1429.60019 |
| [18] | Franchi, B.; Serapioni, R.; Serra Cassano, F., Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321, 3, 479-531, 2001 ·Zbl 1057.49032 |
| [19] | Hajłasz, P.; Koskela, P., Sobolev met Poincaré, Mem. Am. Math. Soc., 145, 688, 2000 ·Zbl 0954.46022 |
| [20] | Heinonen, J.; Koskela, P., Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181, 1, 1-61, 1998 ·Zbl 0915.30018 |
| [21] | Heinonen, J.; Koskela, P.; Shanmugalingam, N.; Tyson, J. T., Sobolev Spaces on Metric Measure Spaces, New Mathematical Monographs, vol. 27, 2015, Cambridge University Press: Cambridge University Press Cambridge ·Zbl 1332.46001 |
| [22] | Hussain, M.; Simmons, D., A general principle for Hausdorff measure, Proc. Am. Math. Soc., 147, 9, 3897-3904, 2019 ·Zbl 1426.28011 |
| [23] | Järvenpää, E.; Järvenpää, M.; Rogovin, K.; Rogovin, S.; Shanmugalingam, N., Measurability of equivalence classes and MEC_p-property in metric spaces, Rev. Mat. Iberoam., 23, 3, 811-830, 2007 ·Zbl 1146.28001 |
| [24] | Kinnunen, J.; Korte, R.; Shanmugalingam, N.; Tuominen, H., A characterization of Newtonian functions with zero boundary values, Calc. Var. Partial Differ. Equ., 43, 3-4, 507-528, 2012 ·Zbl 1238.31008 |
| [25] | Koivusalo, H.; Rams, M., Mass transference principle: from balls to arbitrary shapes, Int. Math. Res. Not., 8, 6315-6330, 2021 ·Zbl 1485.11114 |
| [26] | Korte, R.; Lahti, P., Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 31, 1, 129-154, 2014 ·Zbl 1285.28003 |
| [27] | Lahti, P., Federer’s characterization of sets of finite perimeter in metric spaces, Anal. PDE, 13, 5, 1501-1519, 2020 ·Zbl 1451.30121 |
| [28] | Lahti, P., A new Federer-type characterization of sets of finite perimeter, Arch. Ration. Mech. Anal., 236, 2, 801-838, 2020 ·Zbl 1434.49033 |
| [29] | Lahti, P.; Shanmugalingam, N., Fine properties and a notion of quasicontinuity for BV functions on metric spaces, J. Math. Pures Appl. (9), 107, 2, 150-182, 2017 ·Zbl 1359.30055 |
| [30] | Miranda, M., Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. (9), 82, 8, 975-1004, 2003 ·Zbl 1109.46030 |
| [31] | Negreira, F.; Sequeira, E., Sets with large intersection properties in metric spaces, J. Math. Anal. Appl., 511, 1, Article 126064 pp., 2022 ·Zbl 1495.11087 |
| [32] | Wang, B.; Wu, J., Mass transference principle from rectangles to rectangles in Diophantine approximation, Math. Ann., 381, 1-2, 243-317, 2021 ·Zbl 1483.11150 |
| [33] | Wang, B.-W.; Wu, J.; Xu, J., Mass transference principle for limsup sets generated by rectangles, Math. Proc. Camb. Philos. Soc., 158, 3, 419-437, 2015 ·Zbl 1371.11119 |
| [34] | Zhong, W., Mass transference principle: from balls to arbitrary shapes: measure theory, J. Math. Anal. Appl., 495, 1, Article 124691 pp., 2021 ·Zbl 1460.28009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
