Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2014-11962-9
Published electronically: April 24, 2014
MathSciNet review: 3209322
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] A. S. Besicovitch, On the definition of tangents to sets of infinite linear measure, Proc. Cambridge Philos. Soc. 52 (1956), 20-29. MR 0074496 (17,595d)
  • [2] Carlos A. Cabrelli, Kathryn E. Hare, and Ursula M. Molter, Classifying Cantor sets by their fractal dimensions, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3965-3974. MR 2679618 (2011k:28011), https://doi.org/10.1090/S0002-9939-2010-10396-9
  • [3] Thomas Duquesne, The packing measure of the range of super-Brownian motion, Ann. Probab. 37 (2009), no. 6, 2431-2458. MR 2573563 (2011b:28014), https://doi.org/10.1214/09-AOP468
  • [4] Kenneth Falconer, Fractal geometry, Mathematical foundations and applications, John Wiley & Sons Ltd., Chichester, 1990. MR 1102677 (92j:28008)
  • [5] H. Haase, Non-$ \sigma $-finite sets for packing measure, Mathematika 33 (1986), no. 1, 129-136. MR 859505 (88a:28003), https://doi.org/10.1112/S0025579300013942
  • [6] Kathryn Hare, Franklin Mendivil, and Leandro Zuberman.
    The sizes of rearrangements of cantor sets.
    Canad. Math. Bull. 56 (2013), no. 2, 354-365. MR 3043062
  • [7] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR 1333890 (96h:28006)
  • [8] Peter Mörters and Narn-Rueih Shieh, The exact packing measure of Brownian double points, Probab. Theory Related Fields 143 (2009), no. 1-2, 113-136. MR 2449125 (2009m:60192), https://doi.org/10.1007/s00440-007-0122-x
  • [9] C. A. Rogers, Hausdorff measures, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1970 original, with a foreword by K. J. Falconer. MR 1692618 (2000b:28009)
  • [10] S. James Taylor and Claude Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), no. 2, 679-699. MR 776398 (87a:28002), https://doi.org/10.2307/1999958

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 28A78, 28A80

Retrieve articles in all journals with MSC (2010): 28A78, 28A80


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

American Mathematical Society