Hausdorff Dimension Of Subsets Of Sierpinski Carpet With Linear Structure

This paper discussed two classes of subsets of the general Sierpinski carpets for which the digits in the expansions lie in some groups or in some horizontal fibers with linear frequencies.There are two problems,the first is the fact that its natural covering under the self-affine transformation is not a efficient covering, the second is the fact that the limiting frequency of some digits or fibers may not exit.However.We obtain its Hausdorff dimension spectrum by density theorem. Especially,we find two classes of dense subsets which have the explicit formula of Hausdorff dimension.Meanwhile,we discuss the properties of the corresponding Hausdorff measure.Furthermore,we prove that the Hausdorff measures in their dimensions are infinite if the conditions foil to hold.