### On a Zero-Infinity Law of Olsen

Enrico Zoli
Source: Real Anal. Exchange Volume 34, Number 1 (2008), 215-218.

#### Abstract

Let $\mu$ be a translation-invariant metric measure on $\mathbb{R}$ with the following scaling property: for every $\lambda \in (0,1)$ there exists $b(\lambda)>\lambda$ with $\mu(\lambda X)\geq b(\lambda) \mu(X)$ for all $X \subseteq \mathbb{R}$. If $X$ is a $\mathbb{Z}$-invariant subset of $\mathbb{R}$ with $X/q \subseteq X$ for some $q\in \mathbb{N}\setminus \{1\}$, then $\mu(X)=0$ or $\mu(X\cap O)=\infty$ for every non-empty open set $O$. This refines an earlier result by Olsen.

First Page:
Primary Subjects: 28A12, 28A78
Secondary Subjects: 11J83
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.rae/1242738932
Zentralblatt MATH identifier: 05578226
Mathematical Reviews number (MathSciNet): MR2527134

### References

Y. Bugeaud, M.M. Dodson, and S. Kristensen, Zero–infinity laws in Diophantine approximation, Quart. J. Math. 56 (2005), 311–320.
Mathematical Reviews (MathSciNet): MR2161245
Zentralblatt MATH: 1204.11117
Digital Object Identifier: doi:10.1093/qmath/hah043
M. Csörnyei and R.D. Mauldin, Scaling properties of Hausdorff and packing measures, Math. Ann. 319 (2001), 817–836.
Mathematical Reviews (MathSciNet): MR1825410
Zentralblatt MATH: 0985.28007
Digital Object Identifier: doi:10.1007/PL00004461
M. Elekes and T. Keleti, Borel sets which are null or non-$\sigma$-finite for every translation invariant measure, Adv. Math. 201 (2006), 102–115.
Mathematical Reviews (MathSciNet): MR2204751
Zentralblatt MATH: 1110.28011
Digital Object Identifier: doi:10.1016/j.aim.2004.11.009
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, London, 1995.
Mathematical Reviews (MathSciNet): MR1333890
R.D. Mauldin and S.C. Williams, Scaling Hausdorff measures, Mathematika 36 (1989), 325–333.
Mathematical Reviews (MathSciNet): MR1045791
Digital Object Identifier: doi:10.1112/S0025579300013164
L. Olsen, On the dimensionlessness of invariant sets, Glasgow Math. J. 45 (2003), 539–543.
Mathematical Reviews (MathSciNet): MR2005353
Zentralblatt MATH: 1036.28009
Digital Object Identifier: doi:10.1017/S0017089503001447
H. Weber and E. Zoli, On a theorem of Volkmann, Real Anal. Exchange 31(1) (2006), 1–6.
Mathematical Reviews (MathSciNet): MR2218183
Zentralblatt MATH: 1107.28004
Project Euclid: euclid.rae/1149516813