Cantor sets in higher dimensions II: Optimal dimension constraint for stable intersections
It is well known that a pair of compact sets in $\mathbb{R}^d$ ($d \in \mathbb{N}$) can be separated by small deformations if the sum of their upper box dimensions is less than $d$. In this paper, we demonstrate that this dimension constraint is optimal for regular Cantor sets. Specifically, for any prescribed upper box dimensions whose sum is greater than $d$, we construct classes of pairs of regular Cantor sets that exhibit $C^{1+α}$-stable intersections. Our method is geometrically flexible, enabling the construction of examples with arbitrarily small thickness in both projectively hyperbolic and nearly conformal regimes. These results also extend to the complex setting for holomorphic Cantor sets in $\mathbb{C}^d$. The proof relies on the "covering criterion" for stable intersection introduced in the first part of this series [NZ25], which generalizes the "recurrent compact set criterion" of Moreira-Yoccoz to higher dimensions.