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

http://dx.doi.org/10.4153/CJM-1999-047-4
Canad. J. Math. 51(1999), 1073-1088

  Published:1999-10-01
 Printed: Oct 1999

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.