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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Packing dimension and Cartesian products

Authors: Christopher J. Bishop and Yuval Peres
Journal: Trans. Amer. Math. Soc. 348 (1996), 4433-4445
MSC (1991): Primary 28A80
MathSciNet review: 1376540
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Abstract: We show that for any analytic set $A$ in $\mathbf {R}^d$, its packing dimension $\dim _{\mathrm {P}}(A)$ can be represented as $ \; \sup _B \{ \dim _{\mathrm {H}} (A \times B) -\dim _{\mathrm {H}}(B) \} \, , \, $, where the supremum is over all compact sets $B$ in $\mathbf {R}^d$, and $\dim _{\mathrm {H}}$ denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if $\dim _{\mathrm {P}} (A) < d$. In contrast, we show that the dual quantity $ \; \inf _B \{ \dim _{\mathrm {P}}(A \times B) -\dim _{\mathrm {P}}(B) \} \, , \, $, is at least the ``lower packing dimension'' of $A$, but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)

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Additional Information

Christopher J. Bishop
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651

Yuval Peres
Affiliation: Department of Statistics, University of California, Berkeley, California 94720
Address at time of publication: Institute of Mathematics, The Hebrew University, Givat Ram, Jerusalem 91904, Israel

Keywords: Hausdorff dimension, packing dimension, Cartesian product, tree
Received by editor(s): April 27, 1995
Additional Notes: Supported in part by NSF grant # DMS 9204092 and by an Alfred P. Sloan Foundation Fellowship
Research partially supported by NSF grant # DMS-9404391
Article copyright: © Copyright 1996 American Mathematical Society