On the intersection of Cantor sets with the unit circle and some sequences

For (Formula presented.), let (Formula presented.) be the self-similar set in (Formula presented.) generated by the iterated function system (Formula presented.). In this paper, we investigate the intersection of the unit circle (Formula presented.) with the Cartesian product (Formula presented.). We prove that for (Formula presented.), the intersection is trivial, that is, (Formula presented.) If (Formula presented.), then the intersection (Formula presented.) is nontrivial. In particular, if (Formula presented.), the intersection (Formula presented.) is of cardinality continuum. Furthermore, the bound (Formula presented.) is sharp: there exists a sequence (Formula presented.) with (Formula presented.) such that (Formula presented.) is nontrivial for all (Formula presented.). This result provides a negative answer to a problem posed by Yu (2023). Our methods extend beyond the unit circle and remain effective for many nonlinear curves. We also characterize the intersection of missing digits Cantor sets with the sequence (Formula presented.) by utilizing the Legendre symbol.