Less than $2^{\omega}$ many translates of a compact nullset may cover the real line

TL;DR

The paper constructs a specific measure-zero compact set that, when translated, can intersect every perfect set uncountably, and demonstrates under certain set-theoretic assumptions that fewer than continuum many such translates can cover the real line.

Contribution

It introduces a measure-zero compact set with universal intersection properties and shows the consistency of covering the real line with fewer than continuum many translates.

Findings

Existence of a measure-zero compact set intersecting all perfect sets uncountably

Under certain set-theoretic assumptions, fewer than continuum translates can cover \\RR

Answers to questions by Darji, Keleti, and Gruenhage are provided

Abstract

We answer a question of Darji and Keleti by proving that there exists a compact set $C_0\subset\RR$ of measure zero such that for every perfect set $P\subset\RR$ there exists $x\in\RR$ such that $(C_0+x)\cap P$ is uncountable. Using this $C_0$ we answer a question of Gruenhage by showing that it is consistent with $ZFC$ (as it follows e.g. from $\textrm{cof}(\iN)<2^\omega$) that less than $2^\omega$ many translates of a compact set of measure zero can cover $\RR$.