# A Survey on Newhouse Thickness, Fractal Intersections and Patterns

## Abstract

**:**

## 1. Newhouse’s Thickness

**Observation**

**1.**

**Observation**

**2.**

**Lemma**

**1.**

**Proof.**

**Observation**

**3.**

**Observation**

**4.**

**Example**

**1.**

## 2. The Gap Lemma

#### 2.1. Why Thickness and the Gap Lemma?

- Their convex hulls are disjoint:

- One of the sets is contained in a gap of the other set:

- The sets are “interleaved”, as shown below:

**Lemma**

**2.**

#### 2.2. The Gap Lemma

**Theorem**

**1.**

- 1.
- $conv\left({C}^{1}\right)\cap conv\left({C}^{2}\right)\ne \varnothing $;
- 2.
- Neither set lies in a gap of the other compact set;
- 3.
- $\tau \left({C}^{1}\right)\tau \left({C}^{2}\right)\ge 1$.

**Observation**

**5**(Sharpness of Theorem 1).

**Definition**

**1.**

**Definition**

**2.**

**Proof**

**of**

**Theorem**

**1.**

- Any endpoint of the convex hull of ${C}^{i}$ or a gap of ${C}^{i}$ belongs to ${C}^{i}$.
- If a point belongs to $conv\left({C}^{i}\right)$ but does not belong to ${C}^{i}$, then it is in a gap ${G}_{m}^{i}$.
- If a point belongs to ${C}^{i}$, then (by the assumption ${C}^{1}\cap {C}^{2}=\varnothing $) the point is either outside of $conv\left({C}^{j}\right)$ ($j:=3-i$) or in a gap ${G}_{n}^{j}$.

**First step**. By the first assumption (Equation (1)) and symmetry, we may assume that there is an endpoint of $conv\left({C}^{2}\right)$ that belongs to $conv\left({C}^{1}\right)$, and thus it is in ${C}^{2}\cap conv\left({C}^{1}\right)$. However, since ${C}^{1}\cap {C}^{2}=\varnothing $, then it belongs to ${G}_{{m}_{1}}^{1}\cap {C}^{2}$ for some gap ${G}_{{m}_{1}}^{1}$. Thus, we have

**The inductive step**. Assume that ${G}_{{m}_{i}}^{1}$ and ${G}_{{n}_{i}}^{2}$ are linked gaps, where $B\left({v}_{{n}_{i}}^{2}\right)\subseteq {G}_{{m}_{i}}^{1}$ and ${v}_{{n}_{i}}^{2}$ is an endpoint of ${G}_{{n}_{i}}^{2}$ (the symmetric condition is identical). Let ${u}_{{m}_{i}}^{1}$ be the endpoint of ${G}_{{m}_{i}}^{1}$ that is in ${G}_{{n}_{i}}^{2}$.

## 3. Connection to the Hausdorff Dimension

**Definition**

**3.**

**Hausdorff dimension**of E is the supremum of all real-valued s, for which the s-dimensional Hausdorff content of E is positive (see Figure 5).

**Theorem**

**2.**

**Observation**

**6**(Sharpness of Theorem 2).

**Observation**

**7.**

**Proof**

**of**

**Theorem**

**2.**

- $x\ge 0$ and $y\ge 0$;
- $x+y\le 1$ (because $U\left(L\right)\u228dU\left(R\right)\subseteq A$);

**Claim:**

**Observation**

**8.**

## 4. Thickness and Patterns in Fractals

**Definition**

**4.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Claim: $A+B=[{a}_{2},1+{a}_{1}]$**

- $-A$ is not contained in a gap of $B-t$. This is true because, since $\tau \left(C\right)\ge 1$, $|conv(-A\left)\right|$$\ge \left|G\right|\ge |\mathrm{any}\phantom{\rule{4.pt}{0ex}}\mathrm{gap}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}B-t|$. Analogously, $B-t$ is not contained in a gap of A.
- $conv(-A)\cap conv(B-t)\ne \varnothing $, since we are considering values of $t\in [{a}_{2},1+{a}_{1}]$.
- $\tau (-A)\tau (B-t)\ge 1$.

**Claim: $C\cap \frac{A+B}{2}\ne \varnothing $**

- We have ${a}_{1}=|conv\left(A\right)|\ge \left|G\right|={a}_{2}-{a}_{1}$ such that $2{a}_{1}\ge {a}_{2}$.
- We are assuming that ${a}_{1}\le 1-{a}_{2}$.
- $0<{a}_{1}<{a}_{2}<1$.

**Lemma**

**5.**

**Proof.**

**Theorem**

**3**(Broderick, Fishman and Simmons).

**zoom in**, and Alice, who decides what to

**erase**there. Bob has limits on how far to zoom in, and Alice has limits on how much to erase. There are also special sets called

**winning sets**, which are subsets of the “board game”. A set W is winning if Alice has a strategy guaranteeing that if she does not erase the limit point of convergence for Bob’s moves during the game, then that point belongs to W. Being a winning set (for certain parameters) can be considered another notion of a “large size” for the set.

**Definition**

**5**(Potential game in $\mathbb{R}$).

- For each $m\in {\mathbb{N}}_{0}$, Bob plays first, and then Alice plays.
- On the mth turn, Bob plays a closed ball ${B}_{m}:=B[{x}_{m},{\rho}_{m}]$. The first ball must satisfy ${\rho}_{0}\ge \rho $. The following moves must satisfy ${\rho}_{m}\ge \beta {\rho}_{m-1}$ and ${B}_{m}\subseteq {B}_{m-1}$ for every $m\in \mathbb{N}$.
- On the mth turn, Alice responds by choosing and erasing a finite or countably infinite collection ${\mathcal{A}}_{m}={\left\{{A}_{{\rho}_{i,m}}\right\}}_{i}$ of balls with radii ${\rho}_{i,m}>0$. Alice’s collection must satisfy the following:$$\begin{array}{cc}\hfill \sum _{i}{\rho}_{i,m}^{c}\le {\left(\alpha {\rho}_{m}\right)}^{c}& \phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}c>0\hfill \\ \hfill {\rho}_{1,m}\le \alpha {\rho}_{m}& \phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}c=0\phantom{\rule{4.pt}{0ex}}(\mathit{in}\phantom{\rule{4.pt}{0ex}}\mathit{this}\phantom{\rule{4.pt}{0ex}}\mathit{case},\phantom{\rule{4.pt}{0ex}}\mathit{Alice}\phantom{\rule{4.pt}{0ex}}\mathit{can}\phantom{\rule{4.pt}{0ex}}\mathit{erase}\phantom{\rule{4.pt}{0ex}}\mathit{just}\phantom{\rule{4.pt}{0ex}}\mathit{one}\phantom{\rule{4.pt}{0ex}}\mathit{set}).\hfill \end{array}$$
- Alice is not allowed to erase any set or, equivalently, to pass her turn.
- Bob must ensure that ${lim}_{m\to \infty}{\rho}_{m}=0$.

**outcome of the game**.

**winning set**if Alice has a strategy guaranteeing that

**Lemma**

**6**(Countable intersection property).

**Lemma**

**8**(Invariance under similarities).

**Theorem**

**4.**

**Proposition**

**2.**

**Proof.**

**Alice’s strategy**: If there exists $n\in \mathbb{N}$ such that B intersects ${G}_{n}$ and $\left|B\right|\le min\left\{\right|{L}_{n}|,|{R}_{n}\left|\right\}$, then Alice erases ${G}_{n}$ if it is a legal movement. In any other case (if B does not intersect any gap of S or if $\left|B\right|>min\left\{\right|{L}_{n}|,|{R}_{n}\left|\right\}$), Alice does not erase anything.

**first**gap intersecting ${B}_{{m}_{n}}$, the gap ${G}_{n}$ is uniquely defined (there are not two gaps that Alice should erase in the same turn). In conclusion, it is legal for Alice to erase ${G}_{n}$ in the ${m}_{n}$th turn, and her strategy specifies that she does so. □

## 5. Extensions of Thickness to Higher Dimensions

#### 5.1. Thickness in ${\mathbb{R}}^{d}$ (Useful for the Cut-out Set Type)

#### 5.2. Thickness in ${\mathbb{R}}^{d}$ (Useful in General, Even for Totally Disconnected Sets)

- Each ${S}_{I}$ is a cube and contains ${\left\{{S}_{I,j}\right\}}_{1\le j\le {k}_{I}}$. No assumptions are made on the separation of the ${S}_{I,j}$.
- Every infinite word ${i}_{1},{i}_{2},\cdots $ of indices is of the construction$$\underset{n\to +\infty}{lim}rad\left({S}_{{i}_{1},{i}_{2},\cdots ,{i}_{n}}\right)=0.$$

**Definition**

**6**(Thickness of C associated with the system of cubes ${\left\{{S}_{I}\right\}}_{I}$).

**Definition**

**7.**

**corner Cantor set**$C={C}_{\ell ,n}\subseteq ({\mathbb{R}}^{d},{dist}_{\infty})$, as in Figure 11. Let $g=\frac{2-n\ell}{n-1}$. One can check that the thickness is given by $\tau \left(C\right)=\frac{\ell}{g}=\frac{\ell (n-1)}{2-n\ell}$, and the set is $r:=\frac{1}{2}(2\ell +g)=\left(\ell +\frac{2-n\ell}{2(n-1)}\right)$-uniformly dense.

**Theorem**

**5**

- (1)
- $\tau ({C}^{1},{\left\{{S}_{I}^{1}\right\}}_{I})\tau ({C}^{2},{\left\{{S}_{L}^{2}\right\}}_{L})\ge \frac{1}{{(1-2r)}^{2}}$;
- (2)
- ${\left\{{S}_{I}^{1}\right\}}_{I}$ and ${\left\{{S}_{L}^{2}\right\}}_{L}$ are r-uniformly dense;
- (3)
- ${C}^{1}\cap (1-2r){S}_{\varnothing}^{2}\ne \varnothing $ and $rad\left({S}_{\varnothing}^{1}\right)\ge r\phantom{\rule{4pt}{0ex}}rad\left({S}_{\varnothing}^{2}\right)$.

#### 5.3. An Application to Directional Distance Sets

**Conjecture**

**1**(Falconer’s distance conjecture).

**Theorem**

**6**

**Corollary**

**1.**

- $\tau (C,{\left\{{S}_{I}\right\}}_{I})\ge \frac{1}{1-2r}$;
- ${\left\{{S}_{I}\right\}}_{I}$ is r-uniformly dense with respect to C.

**Proof.**

#### 5.4. Patterns in Thick Sets in ${\mathbb{R}}^{d}$

## Funding

## Conflicts of Interest

## References

- Newhouse, S.E. Nondensity of axiom A(a) on S
^{2}. In Global Analysis; American Mathematical Society: Providence, RI, USA, 1970; pp. 191–202. [Google Scholar] - Newhouse, S.E. The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math.
**1979**, 50, 101–151. [Google Scholar] [CrossRef] - Astels, S. Cantor sets and numbers with restricted partial quotients. Trans. Am. Math. Soc.
**2000**, 352, 133–170. [Google Scholar] [CrossRef] - Boone, Z.; Palsson, E.A. A pinned Mattila–Sjölin type theorem for product sets. arXiv
**2022**, arXiv:2210.00675. [Google Scholar] - Falconer, K.; Yavicoli, A. Intersections of thick compact sets in ℝ
^{d}. Math. Z.**2022**, 301, 2291–2315. [Google Scholar] [CrossRef] - Hunt, B.R.; Kan, I.; Yorke, J.A. When Cantor sets intersect thickly. Trans. Am. Math. Soc.
**1993**, 339, 869–888. [Google Scholar] [CrossRef] - McDonald, A.; Taylor, K. Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse gap lemma. arXiv
**2021**, arXiv:2111.09393. [Google Scholar] - Simon, K.; Taylor, K. Interior of sums of planar sets and curves. Math. Proc. Camb. Philos. Soc.
**2020**, 168, 119–148. [Google Scholar] [CrossRef] [Green Version] - Williams, R.F. How big is the intersection of two thick Cantor sets. In Continuum Theory and Dynamical Systems (Arcata, CA, 1989)? American Mathematical Society: Providence, RI, USA, 1991; pp. 163–175. [Google Scholar]
- Yavicoli, A. Patterns in thick compact sets. Israel J. Math.
**2021**, 244, 95–126. [Google Scholar] [CrossRef] - Yavicoli, A. Thickness and a gap lemma in ℝ
^{d}. arXiv**2022**, arXiv:2204.08428. [Google Scholar] - Yu, H. Fractal projections with an application in number theory. Ergod. Theory Dyn. Syst.
**2020**. [Google Scholar] [CrossRef] - Palis, J.; Takens, F. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. In Cambridge Studies in Advanced Mathematics; Cambridge University Press: Cambridge, MA, USA, 1993; Volume 35. [Google Scholar]
- Falconer, K.J. The Geometry of Fractal Sets. Cambridge Tracts in Mathematics; Cambridge University Press: Cambridge, MA, USA, 1985. [Google Scholar]
- Falconer, K. Techniques in Fractal Geometry; John Wiley & Sons, Ltd.: Chichester, UK, 1997. [Google Scholar]
- Falconer, K. Fractal Geometry, 3rd ed.; John Wiley & Sons, Ltd.: Chichester, UK, 2014. [Google Scholar]
- Keleti, T. A 1-dimensional subset of the reals that intersects each of its translates in at most a single point. Real Anal. Exchange
**1998/1999**, 24, 843–844. [Google Scholar] [CrossRef] - Keleti, T. Construction of one-dimensional subsets of the reals not containing similar copies of given patterns. Anal. PDE
**2008**, 1, 29–33. [Google Scholar] [CrossRef] - Maga, P. Full dimensional sets without given patterns. Real Anal. Exchange
**2010/2011**, 36, 79–90. [Google Scholar] [CrossRef] - Máthé, A. Sets of large dimension not containing polynomial configurations. Adv. Math.
**2017**, 316, 691–709. [Google Scholar] [CrossRef] [Green Version] - Yavicoli, A. Large sets avoiding linear patterns. Proc. Am. Math. Soc.
**2021**, 149, 4057–4066. [Google Scholar] [CrossRef] [Green Version] - Sahlsten, T.; Kuca, B.; Orponen, T. On a continuous Sárközy type problem. arXiv
**2022**, arXiv:2110.15065. [Google Scholar] - Chan, V.; Łaba, I.; Pramanik, M. Finite configurations in sparse sets. J. Anal. Math.
**2016**, 128, 289–335. [Google Scholar] [CrossRef] [Green Version] - Henriot, K.; Łaba, I.; Pramanik, M. On polynomial configurations in fractal sets. Anal. PDE
**2016**, 9, 1153–1184. [Google Scholar] [CrossRef] [Green Version] - Łaba, I.; Pramanik, M. Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal.
**2009**, 19, 429–456. [Google Scholar] - Broderick, R.; Fishman, L.; Simmons, D. Quantitative results using variants of Schmidt’s game: Dimension bounds, arithmetic progressions, and more. Acta Arith.
**2019**, 188, 289–316. [Google Scholar] [CrossRef] [Green Version] - Biebler, S. A complex gap lemma. Proc. Am. Math. Soc.
**2020**, 148, 351–364. [Google Scholar] [CrossRef] - Feng, D.-J.; Wu, Y.-F. On arithmetic sums of fractal sets in ℝ
^{d}. J. Lond. Math. Soc.**2021**, 104, 35–65. [Google Scholar] [CrossRef] - Mattila, P.; Sjölin, P. Regularity of distance measures and sets. Math. Nachr.
**1999**, 204, 157–162. [Google Scholar] [CrossRef]

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Yavicoli, A.
A Survey on Newhouse Thickness, Fractal Intersections and Patterns. *Math. Comput. Appl.* **2022**, *27*, 111.
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Yavicoli A.
A Survey on Newhouse Thickness, Fractal Intersections and Patterns. *Mathematical and Computational Applications*. 2022; 27(6):111.
https://doi.org/10.3390/mca27060111

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2022. "A Survey on Newhouse Thickness, Fractal Intersections and Patterns" *Mathematical and Computational Applications* 27, no. 6: 111.
https://doi.org/10.3390/mca27060111