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2009/2010 Almost Isometry-Invariant Sets and Shadings
Keith Neu
Real Anal. Exchange 35(2): 391-402 (2009/2010).


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.


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Keith Neu. "Almost Isometry-Invariant Sets and Shadings." Real Anal. Exchange 35 (2) 391 - 402, 2009/2010.


Published: 2009/2010
First available in Project Euclid: 22 September 2010

zbMATH: 1221.28008
MathSciNet: MR2527130

Primary: 28A12

Keywords: almost invariance , almost isometry-invariant , almost translation-invariant , Archimedean set , Banach measure , difference set , Hamel basis , Lebesgue measure , shade-almost invariance , shading , sum set

Rights: Copyright © 2009 Michigan State University Press

Vol.35 • No. 2 • 2009/2010
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