Selfsimilar sets of zero Hausdorff measure and positive packing measure
 Yuval Peres,
 Károly Simon,
 Boris Solomyak
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We prove that there exist selfsimilar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑ _{0} ^{8} a _{ n }4^{−n } a _{ n }4^{−n } with digits witha _{ n } ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of selfsimilar sets, e.g., for projections of certain planar selfsimilar sets to lines. We establish the Hausdorff measure result using special properties of selfsimilar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections.
 C. Bandt and S. Graf,Selfsimilar sets 7. A characterization of selfsimilar fractals with positive Hausdorff measure, Proceedings of the American Mathematical Society114 (1992), 995–1001. CrossRef
 L. Carleson,Selected Problems on Exceptional Sets, Van Nostrand, New York, 1967.
 M. Denker and M. Urbański,Geometric measures for parabolic rational maps, Ergodic Theory and Dynamical Systems12 (1992), 53–66.
 K. J. Falconer,Dimensions and measures of quasi selfsimilar sets, Proceedings of the American Mathematical Society106 (1989), 543–554. CrossRef
 K. J. Falconer,Fractal Geometry. Mathematical Foundations and Applications, Wiley, New York, 1990.
 K. J. Falconer and J. D. Howroyd,Projection theorems for boxcounting and packing dimensions, Mathematical Proceedings of the Cambridge Philosophical Society119 (1996), 287–295. CrossRef
 W. Feller,An Introduction to Probability Theory and its Applications II, Wiley, New York, 1966.
 J. D. Howroyd,On dimension and on the existence of sets of finite positive Hausdorff measure, Proceedings of the London Mathematical Society (3)70 (1995), 581–604. CrossRef
 J. E. Hutchinson,Fractals and selfsimilarity, Indiana University Mathematics Journal30 (1981), 713–747. CrossRef
 M. Järvenpää,On the upper Minkowski dimension, the packing dimension, and orthogonal projections, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica Dissertationes, no. 99, 1994.
 R. Kaufman,On Hausdorff dimension of projections, Mathematika15 (1968), 153–155. CrossRef
 M. Keane, M. Smorodinsky and B. Solomyak,On the morphology of γexpansions with deleted digits, Transactions of the American Mathematical Society347 (1995), 955–966. CrossRef
 R. Kenyon,Projecting the onedimensional Sierpinski gasket, Israel Journal of Mathematics97 (1997), 221–238. CrossRef
 J. C. Lagarias and Y. Wang,Integral selfaffine tilings in ℝ ^{ n } I, Journal of the London Mathematical Society (2)54 (1996), 161–179.
 P. Mattila,Orthogonal projections, Riesz capacities and Minkowski content, Indiana University Mathematics Journal39 (1990), 185–198. CrossRef
 P. Mattila,Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.
 D. Mauldin and K. Simon,The equivalence of some Bernoulli convolutions to Lebesgue measure, Proceedings of the American Mathematical Society126 (1998), 2733–2736. CrossRef
 C. G. Moreira,Stable intersections of Cantor sets and homoclinic bifurcations, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire13 (1996), 741–781.
 J. Palis and F. Takens,Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1992.
 Y. Peres and B. Solomyak,Absolute continuity of Bernoulli convolutions, a simple proof, Mathematical Research Letters3 (1996), 231–239.
 Y. Peres and B. Solomyak,Selfsimilar measures and intersections of Cantor sets, Transactions of the American Mathematical Society350 (1998), 4065–4087. CrossRef
 M. Pollicott and K. Simon,The Hausdorff dimension of λexpansions with deleted digits, Transactions of the American Mathematical Society347 (1995), 967–983. CrossRef
 A. Schief,Separation properties for selfsimilar sets, Proceedings of the American Mathematical Society122 (1994), 111–115. CrossRef
 B. Solomyak,On the random series Σ±λ ^{ i } (an Erdős problem), Annals of Mathematics (2)142 (1995), 611–625. CrossRef
 B. Solomyak,Measure and dimension of some fractal families, Mathematical Proceedings of the Cambridge Philosophical Society124 (1998), 531–546. CrossRef
 D. Sullivan,Entropy, Hausdorff dimensions old and new, and limit sets of geometrically finite Kleinian groups, Acta Mathematica153 (1984), 259–277. CrossRef
 S. J. Taylor and C. Tricot,Packing measure and its evaluation for a Browninan path, Transactions of the American Mathematical Society288 (1985), 679–699. CrossRef
 Title
 Selfsimilar sets of zero Hausdorff measure and positive packing measure
 Journal

Israel Journal of Mathematics
Volume 117, Issue 1 , pp 353379
 Cover Date
 20001201
 DOI
 10.1007/BF02773577
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Yuval Peres ^{(1)} ^{(2)}
 Károly Simon ^{(3)}
 Boris Solomyak ^{(4)}
 Author Affiliations

 1. Department of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
 2. Department of Statistics, University of California, 94720, Berkeley, CA, USA
 3. Institute of Mathematics, University of Miskolc, H315, MiskolcEgyetemváros, Hungary
 4. Department of Mathematics, University of Washington, Box 354350, 98195, Seattle, WA, USA