The sets of the form $C + mC$, where $m \in 2 (0; 1)$ and$C$ is the classic Cantor ternary set, are described.
@article{240, title = {Linear combinations of the classic Cantor set}, journal = {Tatra Mountains Mathematical Publications}, volume = {55}, year = {2013}, doi = {10.2478/tatra.v56i0.240}, language = {EN}, url = {http://dml.mathdoc.fr/item/240} }
Pawlowicz, Marta. Linear combinations of the classic Cantor set. Tatra Mountains Mathematical Publications, Tome 55 (2013) . doi : 10.2478/tatra.v56i0.240. http://gdmltest.u-ga.fr/item/240/