Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.
@article{bwmeta1.element.doi-10_2478_s11533-012-0113-8,
author = {Peter Maga},
title = {Coverings and dimensions in infinite profinite groups},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {246-253},
zbl = {1261.22005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0113-8}
}
Peter Maga. Coverings and dimensions in infinite profinite groups. Open Mathematics, Tome 11 (2013) pp. 246-253. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0113-8/
[1] Abért M., Less than continuum many translates of a compact nullset may cover any infinite profinite group, J. Group Theory, 2008, 11(4), 545–553 http://dx.doi.org/10.1515/JGT.2008.033 | Zbl 1146.22002
[2] Barnea Y., Shalev A., Hausdorff dimension, pro-p groups and Kac-Moody algebras, Trans. Amer. Math. Soc., 1997, 349(12), 5073–5091 http://dx.doi.org/10.1090/S0002-9947-97-01918-1 | Zbl 0892.20020
[3] Bartoszynski T., Judah H., Set Theory, A.K.Peters, Wellesley, 1995
[4] Darji U.B., Keleti T., Covering ℝ with translates of a compact set, Proc. Amer. Math. Soc., 2003, 131(8), 2593–2596 http://dx.doi.org/10.1090/S0002-9939-02-06773-4 | Zbl 1017.03023
[5] Elekes M., Steprāns J., Less than 2ω many translates of a compact nullset may cover the real line, Fund. Math., 2004, 181(1), 89–96 http://dx.doi.org/10.4064/fm181-1-4 | Zbl 1095.28005
[6] Elekes M., Tóth Á., Covering locally compact groups by less than 2ω many translates of a compact nullset, Fund. Math., 2007, 193(3), 243–257 http://dx.doi.org/10.4064/fm193-3-2 | Zbl 1120.22002
[7] Gruenhage G., Levy R., Covering ωω by special Cantor sets, Comment. Math. Univ. Carolin., 2002, 43(3), 497–509 | Zbl 1072.03028
[8] Máthé A., Covering the real line with translates of a zero-dimensional compact set, Fund. Math., 2011, 213(3), 213–219 http://dx.doi.org/10.4064/fm213-3-2 | Zbl 1230.03079