Cantor sets in higher dimensions II: Optimal dimension constraint for stable intersections

TL;DR

This paper proves the optimality of a dimension constraint for stable intersections of regular Cantor sets in higher dimensions, extending results to complex settings and utilizing a geometric covering criterion.

Contribution

It demonstrates the sharpness of the dimension sum threshold for stable intersections and constructs examples with small thickness in various regimes.

Findings

Dimension sum greater than d allows for C^{1+α}-stable intersections.

Constructs pairs of Cantor sets with arbitrarily small thickness exhibiting stable intersections.

Extends the results to holomorphic Cantor sets in complex spaces.

Abstract

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 $C^{1+\alpha}$-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.