A variety of questions and results on Cantor sets revolved around the Minkowski sums of Cantor sets and the topological structure or Hausdorff dimension of these sumsets.  For example, Shmeling and Schmerkin showed that given an increasing sequence {x_n} bounded by 0 and 1, there exists a Cantor set C such that x_n is the Hausdorff dimension of C added to itself n times. 

Given any integer n, we will provide a construction for a Cantor set with zero logarithmic capacity such that the Cantor set added to itself n times is a single interval, while a sum of any smaller number of copies of that set is still a Cantor set.