Affine Thickness: Generalizations and Applications

• Affine thickness is a quantitative geometric invariant that extends classical Cantor set thickness to self-affine and high-dimensional sets.
• It employs scale-invariant measures and affine gap lemmas to establish rigorous intersection criteria and pattern theorems in fractal geometry.
• The concept has broad applications in dynamical systems, Diophantine geometry, and Boolean function complexity, highlighting its theoretical and algorithmic significance.

Affine thickness is a quantitative geometric invariant that generalizes the classical notion of thickness for Cantor and Cantor-type sets to higher-dimensional, self-affine, and dynamically defined sets. It provides a scale-invariant measure of the “robustness” or “fatness” of totally disconnected sets in both real and abstract settings, with essential roles in intersection theory, pattern occurrence, Diophantine geometry, and the study of winning sets in dynamical games. Unlike classical thickness, which is associated to similarity contractions, affine thickness remains stable under affine mappings, and its refinements admit powerful gap lemmas and pattern theorems in high-dimensional settings, as developed in recent works (Howat, 23 Jan 2026, Yavicoli, 2022, Jiang, 2022).

1. Definitions and Core Concepts

Affine thickness extends the classical Newhouse thickness to $\mathbb{R}^n$ and, more generally, to self-affine and affinely structured sets. Let $B[0,1]\subset\mathbb{R}^n$ denote the closed unit ball, and let $A = \mathrm{diag}(\beta_{11},\ldots,\beta_{nn})$ be a contracting diagonal matrix with $0 < \beta_{jj} < 1$. The size of a bounded set $F\subset B[0,1]$ with respect to $A$ is

$$S_A(F) = \inf\bigl\{\, t>0 \mid \exists z \in\mathbb{R}^n:\, F\subset A^{1/t}(B[0,1]) + z \bigr\}.$$

For a compact set $C\subset B[0,1]$ with at most countably many bounded gaps $\{G_k\}_{k\in J}$ (ordered so $S_A(G_1)\geq S_A(G_2)\geq\ldots$), and with $E$ the unbounded component, define the $A$-gap distance for each $k$ as

$$GD_A(k, C) = \inf\bigl\{ t>0 \mid \exists z\in\mathbb{R}^n : G_k\cap (A^{1/t}(B[0,1]) + z)\neq\emptyset,\ (A^{1/t}(B[0,1])+z)\cap (E\cup \cup_{i<k}G_i)\neq\emptyset \bigr\}.$$

The affine thickness is then

$$\tau_A(C) = \inf_{k\in J} \left[S_A(G_k)^{-1} - GD_A(k,C)^{-1} \right].$$

If there are no bounded gaps, $\tau_A(C)=+\infty$ if $C$ has nonempty interior and $-\infty$ otherwise.

For $A = \beta I$, a similarity, this definition recovers (up to scaling) Falconer–Yavicoli thickness $\tau(C)$ (Yavicoli, 2022), and in dimension $1$ these notions recover Newhouse’s original definition.

2. Relationship to Classical and Falconer–Yavicoli Thickness

Falconer–Yavicoli thickness in dimension $n$ is given (for Euclidean gaps $G_k$) by

$$\tau(C) = \inf_k \frac{d(G_k,\, E \cup \bigcup_{j<k}G_j)}{\operatorname{diam}(G_k)},$$

where $d(\cdot,\cdot)$ denotes the Euclidean distance (Yavicoli, 2022). Affine thickness generalizes this by allowing for non-homothetic (affine rather than similitude) scaling, with a precise correspondence: for $A = \beta I$ and $\tau(C) > 0$, one finds

$\tau_A(C) = -\log_\beta \tau(C).$

Therefore, affine thickness bridges the classical similarity-based thickness theory and the higher-dimensional, fully affine context. This generalization is crucial for capturing the intersection and pattern phenomena of fractal sets under non-uniform contractions (Howat, 23 Jan 2026).

3. Affine Gap Lemmas and Intersection Theory

The main utility of affine thickness lies in its robust gap lemmas, which provide criteria for nonemptiness of intersections between fractal sets. In $\mathbb{R}$, the classical Newhouse gap lemma (Jiang, 2022) establishes that for Cantor sets $C,D$,

$$\tau(C)\,\tau(D) > 1 \implies C\cap D \neq \emptyset$$

provided they are “linked” (i.e., neither lies wholly in a gap of the other). This is preserved under affine maps: for any nonzero $\lambda,\mu$ and any translations, $\tau(\lambda C + \alpha) = \tau(C)$, $\tau(\mu D + \beta) = \tau(D)$ (Jiang, 2022).

Falconer–Yavicoli and Yavicoli (Yavicoli, 2022) extend the gap lemma to $\mathbb{R}^d$ using systems of balls. Under denseness and separation conditions, and with sufficiently large affine thickness, a gap lemma of the form “$\tau_A(C_1) + \tau_A(C_2) > 0$ implies $C_1\cap C_2\neq \emptyset$” holds when the pair is strongly refinable—meaning one can fill certain gaps to achieve a configuration with appropriately linked gaps (Howat, 23 Jan 2026).

A notable result is the construction of explicit counterexamples showing that the naive product form $\tau(C_1)\tau(C_2)>1$ is insufficient for intersection in higher dimensions without additional hypotheses, in contrast to the 1D case (Howat, 23 Jan 2026).

4. Affine Thickness, Winning Sets, and Pattern Theorems

Affine thickness provides sufficient conditions for a set to be “winning” in matrix-potential games, a generalization of Schmidt’s game to affine settings. Specifically, if $\tau_A(C)\notin\{\pm\infty\}$, then $C$ is winning for the matrix-potentials game (with explicit winning parameters) (Howat, 23 Jan 2026).

A major consequence is that thick sets (with large affine thickness) contain large homothetic patterns: If $C$ is thick in this sense, there exists $M$ such that every finite set of size at most $M$ embeds affinely (up to a scale) inside $C$. The quantitative lower bounds on $M$ depend on $\tau_A(C)$ and the contraction parameters. For self-affine carpets with large expansion ratios, $M$ can be extremely large (Howat, 23 Jan 2026, Yavicoli, 2022).

Furthermore, thick sets are potential-game winning sets in the sense of Broderick–Fishman–Simmons, and one obtains lower bounds on the Hausdorff dimension of intersections of countably many thick sets and of the thick set itself.

5. Affine Thickness in Cantor-Type and Self-Affine Sets

For Cantor-type (Moran) sets in $\mathbb{R}$, affine thickness coincides with the invariance of classical thickness under affine maps: $$\tau(\lambda K + \alpha) = \tau(K), \quad \lambda\neq 0.$$ The key affine gap lemma in $\mathbb{R}$ applies as follows: for $C, D \subset \mathbb{R}$, if neither lies entirely in a gap of the other and $\tau(C)\tau(D)>1$, then $C\cap D\neq \emptyset$, independent of affine scaling (Jiang, 2022).

Beyond 1D, illustrative examples include self-affine Sierpiński carpets: for odd $\mathbf{r} = (r_1,\ldots, r_n)\geq 3$, their affine thickness is

$$\tau_A(C_\mathbf{r}) = t^{-1} \min_{1 \leq i \leq n} \log_{r_i}\left(\frac{r_i-1}{2}\right),$$

where $t$ is a contraction parameter per construction stage. For large $r_j$, these carpets are extremely thick and contain all homothetic copies of sets up to high cardinality (Howat, 23 Jan 2026).

6. Algorithmic and Constructive Perspectives

Affine (or algebraic) thickness also features as a complexity measure for Boolean functions, describing the minimal support size over affine changes of basis in the ANF (algebraic normal form). For functions $f : \mathbb{F}_2^n \to \mathbb{F}_2$, the algebraic thickness $\tau(f)$ is

$$\tau(f) = \min_{A \in \mathcal{A}_n} \| f\circ A \|,$$

where $\mathcal{A}_n$ is the group of affine invertible maps and $\|g\|$ counts nonzero monomials (Boyar et al., 2014). Functions of low algebraic thickness must be constant on large affine subspaces (large normality), with efficient algorithms to find such subspaces. The trade-off between algebraic thickness and normality is quantitatively tight: small $\tau(f)$ implies existence of a large affine flat where $f$ is constant, sharply limiting the complexity of such functions.

7. Significance and Applications

Affine thickness provides a powerful, scale-invariant quantifier of geometric and combinatorial largeness in the study of fractal sets, Diophantine problems, and dynamical games on $\mathbb{R}^n$. Key theoretical implications include:

• Sharp intersection and gap criteria for Cantor-type and self-affine sets (Jiang, 2022, Yavicoli, 2022, Howat, 23 Jan 2026).
• Existence of large homothetic patterns in sufficiently thick sets, generalizing classical pattern results (Howat, 23 Jan 2026, Yavicoli, 2022).
• Quantitative control of Hausdorff dimension for intersections and for individual thick sets, both in the potential-game context and independently (Yavicoli, 2022).
• Clarification of the limitations of classical thickness product criteria in higher dimensions and identification of the necessity for more refined conditions such as strong refinability (Howat, 23 Jan 2026).
• Algorithmic inferences in theoretical computer science, especially in the context of Boolean function complexity (Boyar et al., 2014).

A plausible implication is that affine thickness will continue to be central to bridging fractal geometry, additive combinatorics, and dynamical systems in higher dimensions, with possible further applications in arithmetic geometry and fractal tiling theory.