Self-similar subsets of a class of Cantor sets

We study the self-similar subsets of a class of Cantor sets with nice symmetric structure. Our result generalizes the result of Feng, Rao and Wang (2015) [3], which characterizes all the self-similar subsets of the Middle-Third Cantor set (a question raised by P. Mattila in 1998). We simplify a number-theoretical argument in the paper of Feng, Rao and Wang, and make it applicable to a large class of self-similar sets.