Almost Isometry-Invariant Sets and Shadings

Abstract

An almost isometry-invariant set $A\subset \mathbb{R}$ satisfies $|gA\ \Delta \ A| < \mathbf{c}$ for any isometry $g$ acting on $\mathbb{R}$, where ${\bf c}$ is the cardinality of the continuum. A shading is any set $S\subseteq \mathbb{R}$ in which $\frac{\mu(S\cap I)}{\mu(I)}$ has the same constant value for every finite interval $I$, for any Banach measure $\mu$. (A Banach measure is a finitely additive, isometry-invariant extension of the Lebesgue measure to $2^{\mathbb{R}}$.) In this paper we prove several theorems that show how these two types of sets are related. We also prove several sum and difference set results for almost isometry-invariant sets. Finally, we completely solve a problem involving subsets of Archimedean sets first posed by R. Mabry and partially solved by K. Neu.