Determine the intersection {1/n^2} ∩ C for the middle-third Cantor set

Determine the complete description of the set {1/n^2 : n ∈ N} ∩ C, where C denotes the middle-third Cantor set in [0,1]; equivalently, identify all natural numbers n for which 1/n^2 belongs to C.

Background

The authors paper intersections of Cantor sets with curves and sequences. In the discrete direction, they point out that even for very classical sequences the intersection with the middle-third Cantor set C is poorly understood.

Later in the paper they obtain complete classifications for intersections with {1/n^2} in certain prime-base Cantor sets K_{m,D} under arithmetic constraints via the Legendre symbol, and they provide partial information for C (e.g., specific inclusions). However, a full characterization for C itself remains unknown, and the authors explicitly highlight this gap.