### A Few Results on Archimedean Sets

Keith Neu
Source: Real Anal. Exchange Volume 34, Number 1 (2008), 157-170.

#### Abstract

In a 1990 paper by R. Mabry, it is shown that for any constant $a\in \left(0,1\right)$ there exist sets $A$ on the real line with the property that for any bounded interval $I$, $\displaystyle\frac{\mu(A\bigcap I)}{\mu(I)}=a$, where $\mu$ is any Banach measure. \noindent Many of the constructed sets are Archimedean sets, which are sets that satisfy $A+t=A$ for densely many $t\in {\mathbb{R}}$. In that paper it is shown that if $A$ is an arbitrary Archimedean set, then for a fixed, $\mu$, $\displaystyle\frac{\mu(A\bigcap I)}{\mu(I)}$ is constant. (This constant is called the $\mu$-shade of $A$ and is denoted ${\rm sh}_{\mu}A$.) A problem is then proposed: For any Archimedean set $A$, any fixed Banach measure $\mu$, and any number $b$ between $0$ and ${\rm sh}_{\mu}A$, does there exist a subset $B$ of $A$ such that $\displaystyle\frac{\mu(B\bigcap I)}{\mu(I)}=b$ for any bounded interval $I$? In this paper, we partially answer this question. We also derive a lower bound formula for the $\mu$-shade of the difference set of an arbitrary Archimedean set. Finally, we generalize an intersection result from Mabry's original paper.

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Primary Subjects: 28A12
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Permanent link to this document: http://projecteuclid.org/euclid.rae/1242738928
Zentralblatt MATH identifier: 05578222
Mathematical Reviews number (MathSciNet): MR2527130

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