Hausdorff dimension of limsup sets of rectangles in the Heisenberg group
DOI:
https://doi.org/10.7146/math.scand.a-119234Abstract
The almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group is computed in terms of directed singular value functions.
References
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Balogh, Z. M., Durand-Cartagena, E., Fässler, K., Mattila, P., and Tyson, J. T., The effect of projections on dimension in the Heisenberg group, Rev. Mat. Iberoam. 29 (2013), no. 2, 381–432. https://doi.org/10.4171/RMI/725
Balogh, Z. M., Fässler, K., Mattila, P., and Tyson, J. T., Projection and slicing theorems in Heisenberg groups, Adv. Math. 231 (2012), no. 2, 569–604. https://doi.org/10.1016/j.aim.2012.03.037
Balogh, Z. M., Rickly, M., and Serra Cassano, F., Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric, Publ. Mat. 47 (2003), no. 1, 237–259. https://doi.org/10.5565/PUBLMAT_47103_11
Balogh, Z. M. and Tyson, J. T., Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group, Proc. London Math. Soc. (3) 91 (2005), no. 1, 153–183. https://doi.org/10.1112/S0024611504015205
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Balogh, Z. M., Tyson, J. T., and Wildrick, K., Frequency of Sobolev dimension distortion of horizontal subgroups in Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 2, 655–683.
Besicovitch, A. S., On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1935), no. 1, 321–330. https://doi.org/10.1007/BF01448030
Borel, E., Sur les séries de Taylor, Acta Math. 21 (1897), no. 1, 243–247, Lettre adressée à l'éditeur. https://doi.org/10.1007/BF02417979
Cantelli, F., Sulla probabilitá come limite della frequenza, Atti Accad. Naz. Lincei 26 (1917), no. 1, 39–45.
Chousionis, V., Tyson, J., and Urbański, M., Conformal graph directed markov systems on carnot groups, Mem. Amer. Math. Soc., American Mathematical Society, to appear, eprint arXiv:1605.011127.
Durand, A., On randomly placed arcs on the circle, in “Recent developments in fractals and related fields”, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2010, pp. 343–351. https://doi.org/10.1007/978-0-8176-4888-6_22
Eggleston, H. G., The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31–36. https://doi.org/10.1093/qmath/os-20.1.31
Ekström, F., Järvenpää, E., Järvenpää, M., and Suomala, V., Hausdorff dimension of limsup sets of random rectangles in products of regular spaces, Proc. Amer. Math. Soc. 146 (2018), no. 6, 2509–2521. https://doi.org/10.1090/proc/13920
Ekström, F. and Persson, T., Hausdorff dimension of random limsup sets, J. Lond. Math. Soc. (2) 98 (2018), no. 3, 661–686. https://doi.org/10.1112/jlms.12158
Falconer, K. J., Sets with large intersection properties, J. London Math. Soc. (2) 49 (1994), no. 2, 267–280. https://doi.org/10.1112/jlms/49.2.267
Fan, A.-H., Schmeling, J., and Troubetzkoy, S., A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation, Proc. Lond. Math. Soc. (3) 107 (2013), no. 5, 1173–1219. https://doi.org/10.1112/plms/pdt005
Fan, A.-H. and Wu, J., On the covering by small random intervals, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 1, 125–131. https://doi.org/10.1016/S0246-0203(03)00056-6
Fässler, K. and Hovila, R., Improved Hausdorff dimension estimate for vertical projections in the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 459–483.
Feng, D.-J., Järvenpää, E., Järvenpää, M., and Suomala, V., Dimensions of random covering sets in Riemann manifolds, Ann. Probab. 46 (2018), no. 3, 1542–1596. https://doi.org/10.1214/17-AOP1210
Hovila, R., Transversality of isotropic projections, unrectifiability, and Heisenberg groups, Rev. Mat. Iberoam. 30 (2014), no. 2, 463–476. https://doi.org/10.4171/RMI/789
Jarnik, V., Zur Theorie der diophantischen Approximationen, Monatsh. Math. Phys. 39 (1932), no. 1, 403–438. https://doi.org/10.1007/BF01699082
Järvenpää, E., Järvenpää, M., Koivusalo, H., Li, B., and Suomala, V., Hausdorff dimension of affine random covering sets in torus, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1371–1384. https://doi.org/10.1214/13-AIHP556
Järvenpää, E., Järvenpää, M., Koivusalo, H., Li, B., Suomala, V., and Xiao, Y., Hitting probabilities of random covering sets in tori and metric spaces, Electron. J. Probab. 22 (2017), paper no. 1, 18 pp. https://doi.org/10.1214/16-EJP4658
Khintchine, A., Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), no. 1, 706–714. https://doi.org/10.1007/BF01216806
Persson, T., A note on random coverings of tori, Bull. Lond. Math. Soc. 47 (2015), no. 1, 7–12. https://doi.org/10.1112/blms/bdu087
Persson, T., Inhomogeneous potentials, Hausdorff dimension and shrinking targets, Ann. H. Lebesgue 2 (2019), 1–37. https://doi.org/10.1007/s42081-018-0025-3
Persson, T. and Reeve, H. W. J., A Frostman-type lemma for sets with large intersections, and an application to diophantine approximation, Proc. Edinb. Math. Soc. (2) 58 (2015), no. 2, 521–542. https://doi.org/10.1017/S0013091514000066
Seuret, S., Inhomogeneous random coverings of topological Markov shifts, Math. Proc. Cambridge Philos. Soc. 165 (2018), no. 2, 341–357. https://doi.org/10.1017/S0305004117000512
Seuret, S. and Vigneron, F., Multifractal analysis of functions on Heisenberg and Carnot groups, J. Inst. Math. Jussieu 16 (2017), no. 1, 1–38. https://doi.org/10.1017/S1474748015000092
Vandehey, J., Diophantine properties of continued fractions on the Heisenberg group, Int. J. Number Theory 12 (2016), no. 2, 541–560. https://doi.org/10.1142/S1793042116500342
Zheng, C., A shrinking target problem with target at infinity in rank one homogeneous spaces, Monatsh. Math. 189 (2019), no. 3, 549–592. https://doi.org/10.1007/s00605-019-01309-2
Balogh, Z. M., Durand-Cartagena, E., Fässler, K., Mattila, P., and Tyson, J. T., The effect of projections on dimension in the Heisenberg group, Rev. Mat. Iberoam. 29 (2013), no. 2, 381–432. https://doi.org/10.4171/RMI/725
Balogh, Z. M., Fässler, K., Mattila, P., and Tyson, J. T., Projection and slicing theorems in Heisenberg groups, Adv. Math. 231 (2012), no. 2, 569–604. https://doi.org/10.1016/j.aim.2012.03.037
Balogh, Z. M., Rickly, M., and Serra Cassano, F., Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric, Publ. Mat. 47 (2003), no. 1, 237–259. https://doi.org/10.5565/PUBLMAT_47103_11
Balogh, Z. M. and Tyson, J. T., Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group, Proc. London Math. Soc. (3) 91 (2005), no. 1, 153–183. https://doi.org/10.1112/S0024611504015205
Balogh, Z. M., Tyson, J. T., and Warhurst, B., Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups, Adv. Math. 220 (2009), no. 2, 560–619. https://doi.org/10.1016/j.aim.2008.09.018
Balogh, Z. M., Tyson, J. T., and Wildrick, K., Frequency of Sobolev dimension distortion of horizontal subgroups in Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 2, 655–683.
Besicovitch, A. S., On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1935), no. 1, 321–330. https://doi.org/10.1007/BF01448030
Borel, E., Sur les séries de Taylor, Acta Math. 21 (1897), no. 1, 243–247, Lettre adressée à l'éditeur. https://doi.org/10.1007/BF02417979
Cantelli, F., Sulla probabilitá come limite della frequenza, Atti Accad. Naz. Lincei 26 (1917), no. 1, 39–45.
Chousionis, V., Tyson, J., and Urbański, M., Conformal graph directed markov systems on carnot groups, Mem. Amer. Math. Soc., American Mathematical Society, to appear, eprint arXiv:1605.011127.
Durand, A., On randomly placed arcs on the circle, in “Recent developments in fractals and related fields”, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2010, pp. 343–351. https://doi.org/10.1007/978-0-8176-4888-6_22
Eggleston, H. G., The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31–36. https://doi.org/10.1093/qmath/os-20.1.31
Ekström, F., Järvenpää, E., Järvenpää, M., and Suomala, V., Hausdorff dimension of limsup sets of random rectangles in products of regular spaces, Proc. Amer. Math. Soc. 146 (2018), no. 6, 2509–2521. https://doi.org/10.1090/proc/13920
Ekström, F. and Persson, T., Hausdorff dimension of random limsup sets, J. Lond. Math. Soc. (2) 98 (2018), no. 3, 661–686. https://doi.org/10.1112/jlms.12158
Falconer, K. J., Sets with large intersection properties, J. London Math. Soc. (2) 49 (1994), no. 2, 267–280. https://doi.org/10.1112/jlms/49.2.267
Fan, A.-H., Schmeling, J., and Troubetzkoy, S., A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation, Proc. Lond. Math. Soc. (3) 107 (2013), no. 5, 1173–1219. https://doi.org/10.1112/plms/pdt005
Fan, A.-H. and Wu, J., On the covering by small random intervals, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 1, 125–131. https://doi.org/10.1016/S0246-0203(03)00056-6
Fässler, K. and Hovila, R., Improved Hausdorff dimension estimate for vertical projections in the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 459–483.
Feng, D.-J., Järvenpää, E., Järvenpää, M., and Suomala, V., Dimensions of random covering sets in Riemann manifolds, Ann. Probab. 46 (2018), no. 3, 1542–1596. https://doi.org/10.1214/17-AOP1210
Hovila, R., Transversality of isotropic projections, unrectifiability, and Heisenberg groups, Rev. Mat. Iberoam. 30 (2014), no. 2, 463–476. https://doi.org/10.4171/RMI/789
Jarnik, V., Zur Theorie der diophantischen Approximationen, Monatsh. Math. Phys. 39 (1932), no. 1, 403–438. https://doi.org/10.1007/BF01699082
Järvenpää, E., Järvenpää, M., Koivusalo, H., Li, B., and Suomala, V., Hausdorff dimension of affine random covering sets in torus, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1371–1384. https://doi.org/10.1214/13-AIHP556
Järvenpää, E., Järvenpää, M., Koivusalo, H., Li, B., Suomala, V., and Xiao, Y., Hitting probabilities of random covering sets in tori and metric spaces, Electron. J. Probab. 22 (2017), paper no. 1, 18 pp. https://doi.org/10.1214/16-EJP4658
Khintchine, A., Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), no. 1, 706–714. https://doi.org/10.1007/BF01216806
Persson, T., A note on random coverings of tori, Bull. Lond. Math. Soc. 47 (2015), no. 1, 7–12. https://doi.org/10.1112/blms/bdu087
Persson, T., Inhomogeneous potentials, Hausdorff dimension and shrinking targets, Ann. H. Lebesgue 2 (2019), 1–37. https://doi.org/10.1007/s42081-018-0025-3
Persson, T. and Reeve, H. W. J., A Frostman-type lemma for sets with large intersections, and an application to diophantine approximation, Proc. Edinb. Math. Soc. (2) 58 (2015), no. 2, 521–542. https://doi.org/10.1017/S0013091514000066
Seuret, S., Inhomogeneous random coverings of topological Markov shifts, Math. Proc. Cambridge Philos. Soc. 165 (2018), no. 2, 341–357. https://doi.org/10.1017/S0305004117000512
Seuret, S. and Vigneron, F., Multifractal analysis of functions on Heisenberg and Carnot groups, J. Inst. Math. Jussieu 16 (2017), no. 1, 1–38. https://doi.org/10.1017/S1474748015000092
Vandehey, J., Diophantine properties of continued fractions on the Heisenberg group, Int. J. Number Theory 12 (2016), no. 2, 541–560. https://doi.org/10.1142/S1793042116500342
Zheng, C., A shrinking target problem with target at infinity in rank one homogeneous spaces, Monatsh. Math. 189 (2019), no. 3, 549–592. https://doi.org/10.1007/s00605-019-01309-2
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Published
2020-05-06
How to Cite
Ekström, F., Järvenpää, E., & Järvenpää, M. (2020). Hausdorff dimension of limsup sets of rectangles in the Heisenberg group. MATHEMATICA SCANDINAVICA, 126(2), 229–255. https://doi.org/10.7146/math.scand.a-119234
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