On Sums of "Small" Cantor Sets

Abstract

We provide several results showing that sums of ``small'' Cantor sets can turn into an interval or a finite union of intervals. Specifically, we show that 1) There exists a Cantor set $C\subset [0,1]$ of zero logarithmic capacity such that $C+C=[0,2]$. 2) For any $n\ge 2$ there exists a Cantor set $C\subset [0,1]$ of zero logarithmic capacity such that a Minkowski sum of $n$ copies of $C$ is an interval $[0,n]$, while a sum of any smaller number of copies of $C$ is still a Cantor set. 3) For any $n\ge 2$ there exists a collection of Cantor sets $C_1, C_2, \ldots, C_n\subset [0,1]$ of zero lower box-counting dimension such that a Minkowski sum of any $n-1$ sets from this collection is a Cantor set of zero lower box-counting dimension, while the sum of all $n$ sets is an interval $[0,n]$. 4) There is an uncountable family of Cantor sets of zero lower box-counting dimension such that the sum of any two different sets from this family is a finite union of intervals. 5) For each of the properties above, there are examples of Cantor sets with the same property having zero Hausdorff $h$-measure for an arbitrary given gauge function $h$.