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

• Autorzy: Márton Elekes , Árpád Tóth
• Języki publikacji: EN

Abstrakty

EN

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 $C_{EK}$ is the well known compact nullset considered first by Erdős and Kakutani then ℝ can be covered by cof(𝓝) many translates of $C_{EK}$. 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 cof(𝓝) then there exists a compact set C of Haar measure zero such that G can be covered by cof(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.)

Kategorie tematyczne

• 28C10: Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
• 22B05: General properties and structure of LCA groups
• 03E17: Cardinal characteristics of the continuum
• 28E15: Other connections with logic and set theory
• 22D05: General properties and structure of locally compact groups
• 03E35: Consistency and independence results
• 22E15: General properties and structure of real Lie groups

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2007
• Tom: 193
• Numer: 3
• Strony: 243-257

Daty

wydano

2007

Twórcy

autor

Márton Elekes

• Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary

autor

Árpád Tóth

• Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary

Identyfikatory

DOI

10.4064/fm193-3-2