Affine Gap Lemma

• The paper introduces the gap lemma for affine thickness, quantifying nested affine-scaled probe balls that capture both local and global geometric properties.
• Affine thickness extends classical Newhouse and Falconer–Yavicoli concepts to anisotropic, higher-dimensional settings, proving essential for analyzing self-affine fractals and matrix potential games.
• The proof strategy uses recursive linked gap construction and Cauchy convergence to guarantee intersection under BG-linking, even in complex self-affine configurations.

The Gap Lemma for affine thickness governs intersection properties of subsets of the $n$-cube $B[0,1]\subset \mathbb{R}^n$ under anisotropic scaling, generalizing classical (Newhouse) and Falconer–Yavicoli thickness to affine-invariant and higher-dimensional settings. Affine thickness is central to recent advances in the study of self-affine fractals, robust intersection properties, and matrix potential games, encapsulating the structural rigidity necessary for large-scale pattern existence and dimension conservation under intersection.

1. Affine Thickness in $\mathbb{R}^n$: Definition and Structure

Let $C\subset B[0,1]\subset\mathbb{R}^n$ be compact, and equip $\mathbb{R}^n$ with a fixed norm (typically Euclidean or sup-norm). Fix a diagonal affine contraction $A=\operatorname{diag}(\beta_{11},\dots,\beta_{nn})$ with $0<\beta_{jj}<1$. The set of bounded, path-connected components of $\mathbb{R}^n\setminus C$ is denoted as $(G_k)_{k\in J}$, ordered so that "larger" gaps precede smaller ones; $E$ denotes the unbounded (exterior) component.

The fundamental size parameter for a bounded $F\subset B[0,1]$ is

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

The $m$-th gap distance is defined as

$$GD_A(m,C) =\inf\Bigl\{\,t>0: \exists\,z\in\mathbb{R}^n\ \ \bigl[G_m\cap\left(A^{1/t}(B[0,1])+z\right)\neq\varnothing \ \text{and} \ \left(A^{1/t}(B[0,1])+z\right)\cap(E\cup\underset{i<m}{\bigcup}G_i)\neq\varnothing\bigr]\Bigr\},$$

and the affine thickness is

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

with conventions $\tau_A(C)=-\infty$ if $GD_A(k,C)=0$ for any $k$ and $\tau_A(C)=+\infty$ if $C$ has no bounded gaps and $\text{int}(C)\neq\varnothing$ (otherwise $-\infty$).

Affine thickness thus quantifies how deeply one can nest affine-scaled probe balls inside $C$ around each gap, controlling both local and global geometry. Unlike isotropic (homothetic) thickness, this notion is sensitive to anisotropy and is well-adapted to self-affine and non-uniform constructions (Howat, 23 Jan 2026).

2. The Affine Gap Lemma: Statement and Hypotheses

The classical Newhouse Gap Lemma asserts that for $C_1, C_2\subset\mathbb{R}$ compact, neither contained in a gap of the other, if $\tau(C_1)\tau(C_2)>1$, then $C_1\cap C_2\ne\emptyset$. Direct generalizations to higher-dimensions and to Falconer–Yavicoli’s or affine thickness without extra conditions are invalid, as explicitly demonstrated by counterexamples (Howat, 23 Jan 2026).

The correct multidimensional form incorporates the BG-linked or "strong refinability" property:

• Strongly refinable for $A$: There exist compact $\widetilde C_1,\widetilde C_2\subset B[0,1]$ such that every bounded gap of one is either disjoint from or linked with every gap of the other (their closures intersect, but neither is contained in the other), each set has a gap whose boundary meets the other outside its exterior gap, $$\tau_A(\widetilde C_1)+\tau_A(\widetilde C_2)\ge \tau_A(C_1)+\tau_A(C_2)$$, and $$\widetilde C_1\cap\widetilde C_2\subset C_1\cap C_2$$.

Affine Gap Lemma:

Let $C_1,C_2\subset B[0,1]$ be nonempty, compact, and strongly refinable for $A$. If

$\tau_A(C_1) + \tau_A(C_2) > 0,$

then

$C_1 \cap C_2 \neq \varnothing.$

This is sharp for the affine setting and robust to various cut-out and self-affine constructions. The necessity of the strong refinability/BG-linking condition is established by explicit counterexamples in $\mathbb{R}^n$, $n\ge 2$ (Howat, 23 Jan 2026).

3. Proof Strategy and Structural Reductions

The proof proceeds by reduction to BG-linked pairs, followed by a recursive construction of an infinite chain of linked gaps:

• If $(C_1,C_2)$ are strongly refinable, one can replace them with BG-linked pairs $(\widetilde C_1,\widetilde C_2)$ of no smaller total thickness.
• Using the BG-linked property, construct an infinite sequence $\{(G^1_{s_i}, G^2_{t_i})\}_{i\ge 1}$ of linked gaps with either $s_{i+1}=s_i, t_{i+1}>t_i$ or the reverse.
• At each stage, the sequence ensures that at least one sequence $\text{diam}(G^1_{s_i})$ or $\text{diam}(G^2_{t_i})$ tends to zero.
• By a Cauchy argument, $\lim x_i = \lim y_i$ with $x_i\in \partial G^1_{s_i}\subset C_1$, $y_i\in G^1_{s_i}\cap \partial G^2_{t_i}\subset C_2$, and $d(x_i,y_i)\to 0$, yielding a common point in $C_1\cap C_2$.

The strong refinability and positivity of $\tau_A(C_1)+\tau_A(C_2)$ ensure the necessary chain-building and preclude degenerate "jump" situations where a probe-box could cover both a gap of one and leap over all earlier gaps of the other (Howat, 23 Jan 2026).

4. Comparison with Classical and Falconer–Yavicoli Gap Lemmas

Setting	Gap Lemma Statement	Limitations
Newhouse, $\mathbb{R}^1$	$$\tau(C_1)\tau(C_2)>1\implies C_1\cap C_2\neq\emptyset$$	No extra linking needed
Falconer–Yavicoli, $\mathbb{R}^n$	$$\tau(C_1)\tau(C_2)>1\implies C_1\cap C_2\neq\emptyset$$	Fails for $n\ge2$ without linking
Affine ($\tau_A$), $\mathbb{R}^n$	$\tau_A(C_1)+\tau_A(C_2)>0$ and BG-linked $\implies C_1\cap C_2\neq\emptyset$	BG-linking/strong refinability required

For $n\geq2$, naively extending $\tau(C_1)\tau(C_2)>1$ is invalid; the BG-linked or strong refinability condition is essential. The affine case further distinguishes itself by supporting anisotropic scaling and being tightly bound to the structure of self-affine sets and matrix potential games (Howat, 23 Jan 2026).

5. Illustrative Examples and Counterexamples

• Counterexample (Proposition 2 (Howat, 23 Jan 2026)): Off-center annular-shell-like $C_1,C_2\subset B[0,1]\subset\mathbb{R}^n$ with $\tau_A(C_1)+\tau_A(C_2)>0$ yet $C_1\cap C_2=\emptyset$, invalidating the multidimensional Falconer–Yavicoli lemma.
• Self-affine Sierpinski Carpets: For $A=\mathrm{diag}(r^{-t}, r^{-t})$, $t\in(0,1)$, the Sierpinski carpet $C_r$ has $\tau_A(C_r)=t^{-1}\log_r(\frac{r-1}{2})>0$ for $r\ge3$. Such sets are thick for the matrix potential game and contain homothetic copies of all finite planar patterns up to a size $M$ computably dependent on $r$.
• Countable Intersections: For at most countable families $\{C_i\}$ of thick sets, each with $\tau_A(C_i)>-\infty$, suitable summability of scaling exponents yields a non-trivial intersection over all affine images (Howat, 23 Jan 2026).

6. Applications to Patterns, Intersections, and Matrix Potential Games

Affine thickness enables strong pattern-existence results:

• Existence of a homothetic copy of all finite sets up to size $M$ in thick fractals.
• Dimension lower bounds for intersections of countable families of thick sets, underpinning robust intersection theory in anisotropic and self-affine settings.
• Thick sets (for $\tau_A>0$) are winning sets for matrix potential games, with associated pattern theorems deducing that for large enough parameters, resulting sets contain every finite configuration (with the cardinality bound explicit in terms of thickness) (Howat, 23 Jan 2026).

The matrix potential game framework plays a central role in both the intersection and pattern results, unifying Schmidt-game techniques with affine scaling and gap analysis.

7. Further Developments and Open Problems

The affine Gap Lemma establishes the optimal multidimensional criterion for intersection of thick sets under anisotropic scaling, contingent on strong refinability. Open questions revolve around relaxing the BG-linking condition, sharpening dimension bounds for intersection sets, extending the results to more general self-affine or random fractals, and further integration with potential-game theory in higher-dimensional Diophantine approximation and fractal geometry.

Recent advances (Howat, 23 Jan 2026, Falconer et al., 2021, Yavicoli, 2022) suggest that affine thickness, under appropriate linking conditions, provides the correct generalization of Newhouse thickness to $\mathbb{R}^n$ and offers a robust analytic framework for a wide class of fractal intersection and pattern-existence problems.