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The Hausdorff and Packing Dimensions of Some Sets Related to Sierpi\'nski Carpets

Open Access article
 Printed: Oct 1999
  • Ole A. Nielsen
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The Sierpi\'nski carpets first considered by C.~McMullen and later studied by Y.~Peres are modified by insisting that the allowed digits in the expansions occur with prescribed frequencies. This paper (i)~~calculates the Hausdorff, box (or Minkowski), and packing dimensions of the modified Sierpi\'nski carpets and (ii)~~shows that for these sets the Hausdorff and packing measures in their dimension are never zero and gives necessary and sufficient conditions for these measures to be infinite.
MSC Classifications: 28A78, 28A80 show english descriptions Hausdorff and packing measures
Fractals [See also 37Fxx]
28A78 - Hausdorff and packing measures
28A80 - Fractals [See also 37Fxx]

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