Covering locally compact groups by less than $2^{\omega}$ many translates of a compact nullset

Márton Elekes¹ ; Árpád Tóth²

¹ Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences P.O. Box 127, H-1364 Budapest, Hungary
² Department of Analysis Eötvös Loránd University Pázmány Péter sétány 1/c H-1117 Budapest, Hungary

Fundamenta Mathematicae, Tome 193 (2007) no. 3, pp. 243-257

Résumé

Gruenhage asked if it was possible to cover the real line by less than continuum many translates of a compact nullset. Under the Continuum Hypothesis the answer is obviously negative. Elekes and Stepr# mans gave an affirmative answer by showing that if $C_{\rm EK}$ is the well known compact nullset considered first by Erdős and Kakutani then $\mathbb{R}$ can be covered by ${\rm cof}({\cal N})$ many translates of $C_{\rm EK}$. As this set has no analogue in more general groups, it was asked by Elekes and Stepr# mans whether such a result holds for uncountable locally compact Polish groups. In this paper we give an affirmative answer in the abelian case. More precisely, we show that if $G$ is a nondiscrete locally compact abelian group in which every open subgroup is of index at most ${\rm cof}({\cal N})$ then there exists a compact set $C$ of Haar measure zero such that $G$ can be covered by ${\rm cof}({\cal N})$ many translates of $C$. This result, which is optimal in a sense, covers the cases of uncountable compact abelian groups and of nondiscrete separable locally compact abelian groups. We use Pontryagin's duality theory to reduce the problem to three special cases; the circle group, countable products of finite discrete abelian groups, and the groups of $p$-adic integers, and then we solve the problem on these three groups separately.In addition, using representation theory, we reduce the nonabelian case to the classes of Lie groups and profinite groups, and we also settle the problem for Lie groups. (M. Abért recently gave an affirmative answer for profinite groups, so the nonabelian case is also complete.)

Détail

DOI : 10.4064/fm193-3-2

Keywords: gruenhage asked possible cover real line continuum many translates compact nullset under continuum hypothesis answer obviously negative elekes stepr mans gave affirmative answer showing known compact nullset considered first erd kakutani mathbb covered cof cal many translates set has analogue general groups asked elekes stepr mans whether result holds uncountable locally compact polish groups paper affirmative answer abelian precisely nondiscrete locally compact abelian group which every subgroup index cof cal there exists compact set haar measure zero covered cof cal many translates result which optimal sense covers cases uncountable compact abelian groups nondiscrete separable locally compact abelian groups pontryagins duality theory reduce problem three special cases circle group countable products finite discrete abelian groups groups p adic integers solve problem these three groups separately addition using representation theory reduce nonabelian classes lie groups profinite groups settle problem lie groups nbsp recently gave affirmative answer profinite groups nonabelian complete