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

TL;DR

This paper proves that in certain uncountable locally compact abelian and Lie groups, it is possible to cover the entire group with fewer than continuum many translates of a compact nullset, extending previous results from the real line.

Contribution

It establishes the covering property for a broad class of locally compact groups, using duality and representation theory, generalizing earlier results from the real line to more complex groups.

Findings

Covering of uncountable compact abelian groups with fewer than continuum translates.

Extension of the covering property to nondiscrete separable locally compact abelian groups.

Proof of the covering property for Lie groups and profinite groups.

Abstract

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\=ans gave an affirmative answer by showing that if $C_{EK}$ is the well known compact nullset considered first by Erd\H os and Kakutani then $\RR$ can be covered by $\cof(\iN)$ many translates of $C_{EK}$. As this set has no analogue in more general groups, it was posed as an open question 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 $\cof(\iN)$ then there exists a compact set $C$ of Haar measure zero such that $G$ can be covered by $\cof(\iN)$ 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. We remark that M.~Ab\'ert recently gave an affirmative answer for profinite groups.