Packing dimension and cartesian products

Christopher J. Bishop; Yuval Peres

doi:10.1090/s0002-9947-96-01750-3

Packing dimension and cartesian products

Christopher J. Bishop
, Yuval Peres

Mathematics

Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

We show that for any analytic set A in R^d, its packing dimension dim_p(A) can be represented as sup_B{dim_H(A × B) - dim_H(B)} , where the supremum is over all compact sets B in R^d, and dim_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_p (A) < d. In contrast, we show that the dual quantity inf_B{dim_p(A × B) - dim_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.).

Original language	English
Pages (from-to)	4433-4445
Number of pages	13
Journal	Transactions of the American Mathematical Society
Volume	348
Issue number	11
DOIs	https://doi.org/10.1090/s0002-9947-96-01750-3
State	Published - 1996

Keywords

Cartesian product
Hausdorff dimension
Packing dimension
Tree

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10.1090/s0002-9947-96-01750-3

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title = "Packing dimension and cartesian products",

abstract = "We show that for any analytic set A in Rd, its packing dimension dimp(A) can be represented as supB\{dimH(A × B) - dimH(B)\} , where the supremum is over all compact sets B in Rd, and dimH denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if dimp (A) < d. In contrast, we show that the dual quantity infB\{dimp(A × B) - dimp(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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T1 - Packing dimension and cartesian products

AU - Bishop, Christopher J.

AU - Peres, Yuval

PY - 1996

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N2 - We show that for any analytic set A in Rd, its packing dimension dimp(A) can be represented as supB{dimH(A × B) - dimH(B)} , where the supremum is over all compact sets B in Rd, and dimH denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if dimp (A) < d. In contrast, we show that the dual quantity infB{dimp(A × B) - dimp(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.).

AB - We show that for any analytic set A in Rd, its packing dimension dimp(A) can be represented as supB{dimH(A × B) - dimH(B)} , where the supremum is over all compact sets B in Rd, and dimH denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if dimp (A) < d. In contrast, we show that the dual quantity infB{dimp(A × B) - dimp(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.).

KW - Cartesian product

KW - Hausdorff dimension

KW - Packing dimension

KW - Tree

UR - https://www.scopus.com/pages/publications/27744576565

U2 - 10.1090/s0002-9947-96-01750-3

DO - 10.1090/s0002-9947-96-01750-3

M3 - Article

AN - SCOPUS:27744576565

SN - 0002-9947

VL - 348

SP - 4433

EP - 4445

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 11

ER -

Packing dimension and cartesian products

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Keywords

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