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On packing measures and a theorem of Besicovitch


Authors: Ignacio Garcia and Pablo Shmerkin
Journal: Proc. Amer. Math. Soc. 142 (2014), 2661-2669
MSC (2010): Primary 28A78; Secondary 28A80
Published electronically: April 24, 2014
MathSciNet review: 3209322
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Abstract: Let $ \mathcal {H}^h$ be the $ h$-dimensional Hausdorff measure on $ \mathbb{R}^d$. Besicovitch showed that if a set $ E$ is null for $ \mathcal {H}^h$, then it is null for $ \mathcal {H}^g$, for some dimension $ g$ smaller than $ h$. We prove that this is not true for packing measures. Moreover, we consider the corresponding questions for sets of non-$ \sigma $-finite packing measure and for pre-packing measure instead of packing measure.


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Additional Information

Ignacio Garcia
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Mar del Plata, Buenos Aires Province, Argentina
Email: nacholma@gmail.com

Pablo Shmerkin
Affiliation: Department of Mathematics, Faculty of Engineering and Physical Sciences, University of Surrey, Guilford, GU2 7XH, United Kingdom
Address at time of publication: Torcuato Di Tella University, Av. Figeroa Alcorta 7350 (1428), Buenos Aires, Argentina
Email: pshmerkin@utdt.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-11962-9
Keywords: Packing measure
Received by editor(s): May 28, 2012
Received by editor(s) in revised form: July 24, 2012
Published electronically: April 24, 2014
Additional Notes: The first author was partially supported by CAI+D2009 No. 62-310 (Universidad Nacional del Litoral) and E449 (UNMDP)
The second author was partially supported by a Leverhulme Early Career Fellowship and by a Cesar Milstein Grant
The authors thank the referee for helpful comments
Communicated by: Tatiana Toro
Article copyright: © Copyright 2014 American Mathematical Society