Translating measurable sets

TL;DR

This paper establishes conditions under which null sets can be translated within measurable sets in real analysis, revealing new relationships between measure, density, and translation properties.

Contribution

It proves that under certain density conditions, null sets within compact sets can be translated to match sumsets, extending understanding of additive properties of reals.

Findings

Existence of a closed null set N such that N+B=A+B for certain sets A, B.

If B contains translates of all null subsets of A, then a null set N_0 exists with A extbackslash N_0 translatable into B.

Results relate to consistency in additive set theory and measure theory.

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.