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 Hausdorff dimensions is less than $d$. In this paper, we demonstrate that this dimension constraint is optimal for regular Cantor sets. Specifically, for any prescribed Hausdorff 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 \cite{NZ1}, which generalizes the ``recurrent compact set criterion" of Moreira-Yoccoz to higher dimensions.
The arithmetic difference set
of two compact sets K, K ′ ⊂ R d is a central object in fractal geometry and dynamical systems. A fundamental problem in this area is to identify conditions ensuring that K -K ′ has positive Lebesgue measure (or has non-empty interior).
It is well established that for compact sets satisfying the dimensional deficit
the set K -K ′ has zero Lebesgue measure. Consequently, almost all translations K ′ + t are disjoint from K.
The regime where dim H K + dim H K ′ > d is markedly different and considerably more subtle. Here, one expects not only positive measure but potentially the presence of interior points. This aligns with the Palis conjecture [Pal87] regarding the arithmetic sum of regular Cantor sets on the real line. For such sets (including self-affine sets) one considers the stronger notion of stable intersection. This property implies that not only does K -K ′ possess a non-empty interior, but that this intersection persists robustly under small perturbations of the defining dynamical systems. Recall that a regular Cantor set, by definition, is the unique invariant set under the action of expanding smooth maps with restricted to a specific Markov partition. Two regular Cantor sets are C r -close if their generating maps and Markov partitions are close in the C r topology and Hausdorff topology, respectively. For precise definitions we refer to [NZ25].
While this problem is completely understood in one dimension (d = 1), where the Palis conjecture has been established in both the real [MY01] and complex [AMZ25] settings, the higher dimensional case (d ≥ 2) remains largely unexplored. A primary question is whether the dimension constraint is optimal for all d ∈ N. The aim of this paper is to answer this question affirmatively.
Theorem A. Let d ∈ N and p, q ∈ (0, d) with p + q > d. Then there exists a pair of regular Cantor sets (K, K ′ ) in R d with dim H (K) = p and dim H (K ′ ) = q such that the pair (K, K ′ ) has C 1+α -stable intersection (α > 0). Several distinct methods have been developed to verify stable intersections since the late 1960s. In the case d = 1, the classical gap lemma, introduced by Newhouse [New70], provides an elegant sufficient condition known as the thickness test, although its applicability is limited (see also [Bie20] for the one-dimensional complex setting). However, its recent generalization to higher dimensions [Yav22], even under additional geometric assumptions, does not yield stable intersections. In cases where both Cantor sets have sufficiently large dimensions, namely ⌊dim H K⌋+⌊dim H K ′ ⌋ ≥ d where ⌊x⌋ denotes the integer part of x, a fundamentally different approach was carried out by Asaoka [Asa22]. In this regime, the stable geometric properties of blenders allow one to produce examples of two “transversal” blender-type Cantor sets with C 1 -stable intersections. However, as illustrated in Figure 1, a significant gap remains in the literature for sets with smaller individual dimensions.
The proof of Theorem A is based on the renormalization method. It relies on the “covering criterion” introduced in the first paper of this series [NZ25], where we generalize the “recurrent compact set criterion” of Moreira-Yoccoz [MY01] to higher dimensions. A simplified version of the covering criterion for affine Cantor sets (or so-called self-affine sets) is stated in Theorem 4.1.
This allows us to develop a method to obtain ample classes of examples showing several distinct geometrical features.
Theorem B. The Cantor sets in Theorem A can be chosen to be affine, with arbitrarily small thickness, and nearly homothetic to an arbitrary element of SL(d, R). Furthermore, the construction can be adapted to the complex setting for holomorphic Cantor sets in C d .
Here, a regular Cantor K set is (nearly homothetic, resp.) homothetic to a matrix L ∈ SL(d, R) if the derivative of its generating map at every point p ∈ K is equal to (close to, resp.) λ(p) • L for some λ(p) > 0.
Theorem B demonstrates that the covering criterion can be applied to pairs of Cantor sets generated by expanding maps that share a common invariant (strong) unstable cone field. This, in particular, shows how the criterion operates for the class of Cantor sets that are far from being quasi-conformal; that is, it remains effective even in the presence of strong affine distortion and anisotropy. The techniques developed herein suggest that the covering criterion is sufficiently powerful to address the stable intersection problem in the critical case where K = K ′ . We formulate this in the following conjecture.
Conjecture 1.1. Every regular Cantor set K ⊂ R d with dim H (K) > d/2 can be C ∞ -approximated by a regular Cantor set K such that the pair ( K, K) has C r -stable intersection for some r > 1.
This conjecture can be viewed as a “stable” version of Falconer’s distance set conjecture [Fa86] adapted to regular Cantor sets. We remark that Falconer’s conjecture is still ope
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