TITLE: Does interval spacing effect Hausdorff dimension of Cantor set?
QUESTION [5 upvotes]: Let $C=\bigcap_{j=0}^{2^n}C_j$, $C_0=[0,1]$, and the intervals in the construction of each stage of $C_j$ consists of removing the center 1/3 from the $j-1$ stage intervals.  In other words, the Cantor middle third set.  The dimension of this set is known to be $\frac{\log 2}{\log 3}$.  Now consider the following adjustment: we now only require allow the spacing between the intervals at each stage, $j$, to be disjoint. That is, we only require spacing between intervals to be >0. It is clear to me, from the definition of Hausdorff dimension that interval spacing should not affect an upper bound calculation on a Cantor Set. 
My question is this: Can changing the nonnegative spacing between intervals cause the lower bound calculation to diverge from the upper bound calculation in the context of Cantor sets? For sets in general? If yes for the Cantor sets, can someone direct me to a lower bound technique that explicitly incorporates spacing. If yes only for sets in general, what about Cantor sets makes spacing irrelavant?

REPLY [5 votes]: I regret having to note that the other answer is wrong. It is well known, since work of Moran in 1946, that the Hausdorff dimension of the particular set that you describe does not depend on the location of the intervals at each step. For a proof you can see the book Dimension and Recurrence in Hyperbolic Dynamics, by Barreira (see Theorem 3.1.1).