A pinned Mattila-Sjölin type theorem for product sets

Falconer’s distance problem conjectures a relationship between the structure of a set $E$ and the size of its distance set $\Delta(C):=\{|x-y|:x,y\in C\}$. The notion of structure and size used by Falconer is Hausdorff dimension and Lebesgue measure denoted $\dim_{H}(\cdot)$ and $\mathcal{L}(\cdot)$ respectively. In [5], Falconer proved that if $C\subset\mathbb{R}^{d}$ is compact with $\dim_{H}(C)>\frac{d+1}{2}$ then $\mathcal{L}(\Delta(C))>0$. The conjecture is that we can lower the threshold $\frac{d+1}{2}$ to $\frac{d}{2}$. Many improvements have been made on this threshold. When $d\geq 3$, the best known threshold is $\frac{d^{2}}{2d-1}$ which was first achieved by Du, Guth, Ou, Wang, Wilson and Zhang in [2] when $d=3$ and generalized to higher dimensions by Du and Zhang in [4]. Interestingly, this threshold can be improved for even dimensions. When restricting $d$ to even integers and for $d\geq 2$, the current threshold is $\frac{d}{2}+\frac{1}{4}$. For $d=2$, this threshold was obtained by Guth, Iosevich, Ou and Wang in [10] and generalized to higher even dimensions by Du, Iosevich, Ou, Wang and Zhang in [3].

Another version of the Falconer distance problem, the pinned distance problem, has also seen much attention. Instead of looking at all possible distances from any two elements in a set, we fix one element in the set and only look at the distances the exist from this fixed point, i.e. for $x\in C$ the pinned distance set is $\Delta_{x}(C):=\{|x-y|:y\in C\}$. We can then ask what dimensional requirements are needed so that $\mathcal{L}(\Delta_{x}(C))>0$. This is a stronger version of the Falconer distance problem since we can write

so that if $\Delta_{x}(C)$ has positive Lebesgue measure for some $x\in C$, then we automatically get that $\Delta(C)$ has positive Lebesgue measure. This problem was first investigated by Peres and Schlag in [24] where they proved

which, in particular, implies that if $\dim_{H}(E)>\frac{d+1}{2}$ then there exists $x\in E$ such that $\mathcal{L}(\Delta_{x}(E))>0$. Refinements on this problem have been made, particularly when restricted to the plane. For example, Shmerkin in [25] proved that for $E\subset\mathbb{R}^{2}$ one has

when $E$ is a set satisfying $1<\dim_{H}(E)=\dim_{P}(E)$, where $\dim_{P}(\cdot)$ denotes packing dimension. Moreover, this was improved by Keleti and Shmerkin in [15] to the relaxed setting of when $\dim_{P}(E)\leq 2\dim_{H}(E)-1$.

Due to Mattila [19], one method to prove the Falconer distance problem is to show that there exists a measure $\mu$ on $E$ such that

and this approach has been called the Mattila integral approach. Surprisingly, the dimensional thresholds for the pinned and non-pinned versions of the distance problem match when one studies them via the Mattila integral approach. This is due to Liu in [17] where he establishes an identity for $L^{2}$ integrals.

A variant of the distance problem asks to find dimensional conditions on $C$ such that not only will it be true that $\mathcal{L}(\Delta(C))>0$ but that $\Delta(C)$ will contain an interval. Let $(\cdot)^{\circ}$ denote the interior of a set. This problem was also investigated by Peres and Schlag in [24] where they proved a pinned version of this problem. In particular, they showed

which shows that if $\dim_{H}(E)>\frac{d+2}{2}$ then one can find $x\in E$ such that $\Delta_{x}(E)$ contains an interval. Iosevich and Liu in [12] obtained the bound $\frac{(d+1)^{2}}{d-1}-\frac{d+1}{d-1}\dim_{H}(E)$ which improves on Equation (1) when $\dim_{H}(E)>\frac{d+3}{2}$. In the non-pinned setting, Mattila and Sjölin in [18] refined this result when they proved that if $\dim_{H}(C)>\frac{d+1}{2}$ then $\Delta(C)$ contains an interval, now known as the Mattila-Sjölin theorem. Finding when some kind of geometric operation on a set has non-empty interior is an active research area. For example Iosevich, Mourgoglou, and Taylor in [13] generalized the Mattila-Sjölin theorem to the case where the set $\Delta_{B}(E)=\{\|x-y\|_{B}:x,y\in E\}$ has non-empty interior, where $\|\cdot\|_{B}$ is the metric induced by the norm defined by a symmetric bounded convex body $B$ with a smooth boundary and everywhere non-vanishing Gaussian curvature. Improvements on this threshold have been made by Koh, Pham, and Shen in [16] for product sets, i.e. sets of the form $C=E\times\cdots\times E=E^{d}\subset\mathbb{R}^{d}$ where $E\subset\mathbb{R}$ compact.

Another example of this, which serves as the basis of this paper, is Theorem 1.8 in [20] proven by McDonald and Taylor which will be stated below, and it determines the amount of edge lengths one can find in the set $K_{1}\times K_{2}$ when $K_{1},K_{2}\subset\mathbb{R}$ are Cantor sets. Instead of taking Hausdorff dimension to be the structure of the set, McDonald and Taylor take thickness, denoted by $\tau$, to be the structure of their sets. The definition of thickness is defined below.

First introduced by Newhouse in [21], he gave a definition of thickness specifically for Cantor sets on the real line. Newhouse used this notion of thickness to prove that if $K_{1},K_{2}\subset\mathbb{R}$ are Cantor sets such that $\tau(K_{1})\tau(K_{2})>1$ then $K_{1}\cap K_{2}\neq\emptyset$, a result now known as the Newhouse Gap Lemma. Problems of determining elements in distance sets can be converted into problems of determining intersections of sets, so having a notion of structure which allows one to determine when two sets intersect can be of high importance to questions related to distance sets. Falconer and Yavicoli in [6] generalized both the definition of thickness and the Newhouse Gap Lemma to arbitrary compact sets in $\mathbb{R}^{d}$. Before stating the definition we describe the "gaps" of a compact set.

We will write $\operatorname{dist}$ for the usual distance between points of non-empty subsets of $\mathbb{R}^{d}$ and $\operatorname{diam}$ for the diameter of a non-empty subset of $\mathbb{R}^{d}$.

When restricted to $\mathbb{R}$, this definition of thickness is equivalent to one given by Newhouse. With this definition in mind, the kinds of sets that we are considering are ones that look like they have holes which are been punched out of them. Another way of putting it is that the sets we consider need to have some kind of border. Thus, one drawback to this definition is that there are sets which are of importance in geometric measure theory but do not fit into the framework of Definition 1.2. One example is the four corner Cantor set (and its generalizations) because it does not have any bounded gaps. Therefore it should be noted that recently, Yavicoli in [27] gave a different definition of thickness and proved a type of gap lemma which did allow her to study sets such as the four corner Cantor set. Nevertheless, Falconer and Yavicoli in [6] used the definition of thickness given above to give a generalized version of the Newhouse Gap Lemma which is stated as follows:

Before stating the result of McDonald and Taylor, we provide some preliminary definitions on graphs and specifically, trees.

The usual definition of the distance set is a $1$-chain except with the element $0$ removed, i.e. if we make $G$ a $1$-chain then $\Delta(B)=\Delta_{G}(B)\cup\{0\}$. The element $0$ is removed from $G$ distance sets because we require non-degeneracy conditions in some of the later proofs. McDonald and Taylor in [20] used Newhouse’s definition to prove the following theorem:

The goal of this paper is to extend the above theorem to compact sets in $\mathbb{R}^{d}$ satisfying the same thickness condition, and also extending the work to a pinned tree distance set. This theorem uses and extends the work from Simon and Taylor in [26]. In that paper, they also prove a Mattila-Sjölin type theorem by considering thickness instead of Hausdorff dimension. In particular, Corollary 2.12 in [26] states that for $K_{1},K_{2}\subset\mathbb{R}$ Cantor, if $\tau(K_{1})\tau(K_{2})>1$ then

Extensions to pinned tree distance sets for Cantor sets in $\mathbb{R}$ can also be concluded from the work of McDonald and Taylor, which is discussed below.

A key ingredient in proving statements about non-empty interior of tree distance sets is having access to a strong statement about pinned distance sets, which is presented next and is adapted and extended from Lemma 3.5 in [20].

The proof of this theorem is given in Section 5. We note that this result not only states existence of such an $x^{0}$, but rather we can take any $x^{0}\in C_{1}\times C_{2}$ and obtain this result. Furthermore, this theorem gives a Mattila-Sjölin type result in the pinned distance setting. Indeed, since we are taking a neighborhood around $x^{0}$, we immediately obtain the following corollary:

In Section 2 we construct sets $C_{1}\times C_{2}$, where $C_{1},C_{2}\subset\mathbb{R}^{d}$, for which the above theorem can be applied and such that the dimension of $C_{1}\times C_{2}$ approaches $2d-1$. Then note that when $d=1$ (so $C_{1}\times C_{2}\subset\mathbb{R}^{2})$, the threshold of $2d-1$ turns into $1$, which matches the threshold in the Falconer distance conjecture and both the pinned Falconer problem as well as the Mattila-Sjölin variant. This gives some evidence to the conjecture that we should expect the same threshold for the pinned Mattila-Sjölin question. We also point out that, since we are looking at the product of sets, the threshold that comes from Equation (1) is $\frac{2d+2}{2}$. Thus, we can find a sequence of sets with dimension converging to $2d-1$, which falls below $\frac{2d+2}{2}$ for $d=1$ and matches it when $d=2$. However, for $d\geq 3$, the threshold $\frac{2d+2}{2}$ is lower than $2d-1$ (and thus, better).

Before stating the main result, we fix some notation. Let $T^{k}$ denote a tree with $k$ edges, i.e. a tree with $k+1$ vertices. Then we define the pinned tree distance set with $k$ edges by

Using the above theorem, we can extend the work of McDonald and Taylor to the following, which is the main result of this paper:

Note that the requirements for Theorem 1.10 are no more stringent than the requirements for Theorem 1.8. Thus, we again can construct a wide range of sets of the form $C_{1}\times C_{2}\in\mathbb{R}^{2d}$ of with dimension going down to dimension $2d-1$.

To conclude the introduction, we note that there have been numerous other results on establishing the existence of configurations of a set from the perspective of Hausdorff dimension. For example, Palsson and Romero Acosta in [23] establish that when $\dim_{H}(E)>\frac{2}{3}d+1$ for $d\geq 4$, then the set of congruence classes of triangles formed by triples of points in $E$ will have non-empty interior, with further improvements in low dimensions in a very recent preprint [22].

As another variant of the problem, we can describe distances corresponding to a tree as a function $\Phi$, which is a vector-valued function describing the distances between a set of points. There has also been work when allowing $\Phi$ to describe more general operations, and determining when the resulting set under $\Phi$ has non-empty interior. This was done by Greenleaf, Iosevich, and Taylor in [7] where they considered a suitable class of functions $\Phi:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ to establish the resulting 2-point configuration set $\Delta_{\Phi}(E):=\{\Phi(x,y):x,y\in E\}$ has non-empty interior given lower bounds on $\dim_{H}(E)$ which depend on $\Phi$. In [9], the same authors built upon this work to establish similar results when considering more general k-point configurations and in their most recent paper [8] they can even handle triangles matching the result in [23].

The proof of Theorem 1.10 will be done by induction on $k$ (the number of edges in the tree), and heavily relying on Theorem 1.8. Proving Theorem 1.8 will largely follow the same strategy as is done in [20], however there are clear difficulties that arise when going to higher dimensions, and these issues will be discussed throughout this document

Here, we give an example of a sequence of sets, which will be Sierpiński carpets, for which we can apply Corollary 1.10 and for which the dimension of the sets in this sequence converges to $2d-1$. For a more detailed discussion and construction of these sets see [6]. For simplicity, we will assume each function is the iterated function system of the following Sierpiński carpets will have the same contraction ratios. That is, let $n\in\mathbb{N}_{\geq 3}$ be an odd number and set

Then the family of maps $\{f_{\mathbb{i}}:\mathbb{i}\in D\}$, where $f_{\mathbb{i}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and is defined by

forms an iterated function system, so that there exists a unique and non-empty compact $C\subset\mathbb{R}^{d}$ such that $C=\bigcup_{\mathbb{i}\in D}f_{\mathbb{i}}(C)$. Furthermore, $\dim_{H}(C)=s$ where $s$ is the unique number satisfying $\#D\left(\frac{1}{n}\right)^{s}=1$. For verification of these facts, see [11]. This means $\dim_{H}(C)=\frac{\log(\#D)}{\log(n)}$. The set $C$ corresponds to initially breaking up the unit cube $[0,1]^{d}$ into $n^{d}$ boxes and removing the middle one, then iteratively performing this process on the remaining boxes. Since we are only removing the middle one, we have $\#D=n^{d}-1$. Thus, $\dim_{H}(C)=\frac{\log(n^{d}-1)}{\log(n)}$. Furthermore, in [6] it is shown that $\tau(C)=\frac{n-1}{2\sqrt{d}}.$

Another kind of Sierpiński carpet that we can calculate the Haudsorff dimension and thickness of is one in which instead of just removing the middle box, we only keep the "outer boundary" of boxes. That is, for $n\geq 5$ and $n$ odd, we remove the middle $(n-2)^{d}$ squares and keep the boxes on the outer edge, then perform this process iteratively on the remaining squares. For this set, say $C$, we have $\dim_{H}(C)=\frac{\log(n^{d}-(n-2)^{d})}{\log(n)}$. By a similar discussion as found in [6], the thickness is then

Thus, if we take $C_{n}$ to be a Sierpiński carpet of the first kind described, with contraction ratio $1/n$, and $C_{m}$ to be a Sierpiński carpet of the second kind described, with contraction ratio $1/m$. Then if $1<\tau(C_{n})\tau(C_{m})$, this yields $2d<\frac{n-1}{m-2}$. If $n$ is fixed, this inequality is satisfied, for example, by taking $m$ to be largest odd integer such that $m<\frac{n-1+4d}{2d}$. So an $n\rightarrow\infty$, we choose $m_{n}$ to a sequence of the largest odd numbers such that $m_{n}<\frac{n-1+4d}{2d}$. Then $\dim_{H}(C_{n})\rightarrow d$ and $\dim_{H}(C_{m_{n}})\rightarrow d-1$ as $n\rightarrow\infty$. Thus, for any $x\in C_{n}\times C_{m_{n}}$ and any $k\in\mathbb{N}$, the set $\Delta_{T_{x}^{k}}(C_{n}\times C_{m_{n}})$ has non-empty interior and

as $n\rightarrow\infty$.

First, note that Theorem 1.10 immediately establishes the following result:

One can then ask how low can we go, in terms of Hausdorff dimension, to obtain the result of the above theorem? Falconer and Yavicoli in [6] provide two estimates on this subject. The first of these estimates uses the following definitions:

where $d$ is being taken as in $\mathbb{R}^{d}$.

For $C\subset\mathbb{R}^{d}$, since $\dim_{H}(C\times C)\geq 2\dim_{H}(C)$ the lower estimates on $\dim_{H}(C\times C)$ will be of the form $2\left(d-K_{1}\frac{\tau(C)^{-d}}{\beta^{d}|\log(\beta)|}\right)$ and $2\left(d-1+\frac{\log 2}{\log(1+1/\tau(C))}\right)$. For large thickness, the first estimate is lower (and therefore better since we want to know how small we can make the dimension) and for small thickness, the second estimate is lower (and therefore better). But when considering compact sets $C$ such $\tau(C)$ is close to $1$, this restricts us to the second estimate. This is because to fulfill the requirement $\tau(C)^{-c}\leq\frac{1}{K_{2}}\beta^{c}(1-\beta^{d-c})\leq\frac{1}{K_{2}}$, since $K_{2}$ is very large, we need to $\tau(C)$ to be much larger than $1$. Thus, for $\tau(C)>1$ and observing the second estimate, we have

So for $\tau(C)$ close to $1$, we have the Mattila-Sjölin type result of Theorem 3.1 where we allow the dimension of our set to get down to $2\left(d-1+\frac{\log 2}{\log 3}\right)$. Note that this lower bound is worse than what is found in the Mattila-Sjölin theorem, which in this case would be $\frac{2d+1}{2}$, with the only improvement on the bound being the case when $d=1$. However, there has been work done on establishing chain and tree configurations from the point of view of Hausdorff dimension rather than thickness. Bennett, Iosevich, and Taylor in [1] were able to establish the existence of finite chain configurations with the Mattila-Sjölin threshold of $\frac{d+1}{2}$, i.e. the assumption they were placing on sets was that $E\subset\mathbb{R}^{d}$ for $d\geq 2$ and $\dim_{H}(E)>\frac{d+1}{2}$, which is enough to guarantee the existence of chain configurations. Building on this work, Iosevich and Taylor in [14] generalized the result to establish the existence of tree configurations using the Mattila-Sjölin threshold.

We now introduce our distance function, $g_{x,t}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$. For a fixed $x\in\mathbb{R}^{d}$ this function will have the property that for $C_{1},C_{2}\subset\mathbb{R}^{d}$ we will have $t\in\Delta_{x}(C_{1}\times C_{2})$ provided that $C_{2}\cap g_{x,t}(C_{1})\neq\emptyset$. As pointed out in the introduction, problems of distance sets can be turned into problems of finding when sets intersect. This is one variant of that idea. Now fix $(x,t)\in\mathbb{R}^{2d}\times[0,\infty)$ and for $i\in\{1,\cdots,d\}$ define $\widetilde{g}^{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ by

We do this because if we set $y_{i+d}=g^{i}(y_{i})$ then $\frac{1}{d}t^{2}=(y_{i+d}-x_{i+d})^{2}+(y_{i}-x_{i})^{2}$ for all $i$, which then means that by summing each of these terms we will have

or in other words, the distance between $(y_{1},\cdots,y_{2d})$ and $(x_{1},\cdots,x_{2d})$ is $t$. Finally defined $g_{x,t}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ by

Then we do indeed obtain the property that $t\in\Delta_{x}(C_{1}\times C_{2})$ provided $C_{2}\cap g_{x,t}(C_{1})\neq\emptyset$.

Now we list an interesting property of $g_{x,t}$. From here on out, $g_{x,t}^{\prime}$ will denote the Jacobian matrix of $g_{x,t}$. Then for $z=(z_{1},\cdots,z_{d})\in\mathbb{R}^{d}$ we have

Thus, $g^{\prime}_{x,t}(z)$ is a diagonal matrix. Furthermore, the norm that we will be working with for $g^{\prime}_{x,t}(z)$ is the operator norm which is the square root of the largest eigenvalue of $g^{\prime}_{x,t}(z)^{T}g^{\prime}_{x,t}(z)$ where $T$ denotes the transpose. Then because $g^{\prime}_{x,t}(z)$ is diagonal, we have $g^{\prime}_{x,t}(z)^{T}=g^{\prime}_{x,t}(z)$ and therefore

so that

This estimate will come in later. The function $g_{x,t}$ satisfies another useful property.

We start this section with a lemma that allows us to write $\tau$ in a, sometimes, more convenient way.

Throughout the proof of Theorem 1.8 and Lemma 5.11, it will be convenient for our pinned point to be the origin and for us to be able work with boundary points of gaps of $C_{1}$ and $C_{2}$ to exist on the diagonal. This seems to be, initially, a highly restrictive condition to put on our sets $C_{1}$ and $C_{2}$. Fortunately though, we can do this by utilizing the fact that affine transformations are isometries and that thickness is preserved under affine transformations. The next lemma characterizes this shift.