Translating measurable sets

TL;DR

The paper investigates when the property that every null subset of a measurable set $A$ can be translated into another set $B$ implies that $A$ itself can be translated into $B$. It shows this implication fails in general, but yields positive results under regularity conditions on $B$ (such as $d$-closed and $G_\delta$ in the Euclidean topology) or in the compact case with positive upper density of $B$, where one can extract a null $N$ with $N+B=A+B$ and, for compact $A,B$, a closed null $N\subset A$ with $N+B=A+B$. The proofs combine density-topology arguments, Vitali-covering constructions to obtain compact pieces with positive upper density inside $B$, and minimal-element arguments to produce null witnesses, linking the results to additive properties of the reals. The work also provides a sharp example showing the limits of the results and discusses related consistency questions and potential extensions within the density-topology framework.

Abstract

We prove that if $A,B$ are compact subsets of $\mathbb{R}$ such that the upper density of $B$ is positive at every point of $B$, then there is a closed null set $N\subset A$ such that $N+B=A+B$. As a corollary we find that if $A,B\subset \mathbb{R}$ are measurable, and every null subset $N$ of $A$ can be translated into $B$ (that is, if $B$ contains a suitable translate of $N$), then there is a null set $N_0$ such that $A\setminus N_0$ can be translated into $B$. The topic is related to some consistency results of the theory of additive properties of the reals.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Theorem 3.1