Fat Cantor sets and their skinny companions

The terms fat and skinny in the title are vernacular references to Cantor sets of positive and zero measure respectively. The paper demonstrates that a fat Cantor subset of $$, $$, possesses a skinny companion that forms a Cantor subset of $$ where $$ is the total length of all the gaps associated with the ternary construction of the fat Cantor set. Moreover, elements of the fat Cantor set can be decomposed and expressed as the sum of two components. One of the components is an element of $$. The other component is an element of the skinny companion contained in $$.