Equilateral triangles in subsets of of large Hausdorff dimension
Abstract.
We prove that subsets of , of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles in two dimensions (see [3]). We show that no extra conditions are needed in dimensions four and higher. The three dimensional case remains open.
Some interesting parallels exist between the triangle problem in Euclidean space and its counterpart in vector spaces over finite fields. We shall outline these similarities in hopes of eventually achieving a comprehensive understanding of this phenomenon in the setting of locally compact abelian groups.
1. Introduction
An old and classical problem that arises in different forms in geometric combinatorics, geometric measure theory, ergodic theory and other areas is to show that a sufficiently large set contains vertices of a given geometric configuration. In the realm of positive Lebesgue density, this idea can be encapsulated in the following theorem due to Tamar Ziegler, building on previous results due to Bourgain, Furstenberg, Katznelson, Weiss and others (see, for example, [2] and [5]).
Theorem 1.1.
(T. Ziegler ([11]) Let , of positive upper Lebesgue density in the sense that , where denotes the dimensional Lebesgue measure. Let denote the neighborhood of . Let , where is a positive integer. Then there exists such that for any and any there exists congruent to .
This result nearly settles the issue of simplexes in sets of positive upper Lebesgue density, though even there an interesting open question of whether the neighborhood fudge factor in Theorem 1.1 can be eliminated in the case, say, of nondegenerate triangles. If the triangle is allowed to be degenerate, an example due to Bourgain ([2]) shows that the result is not in general true without the fudge factor.
A natural question that arises at this point is whether a compact set in of Hausdorff dimension contains a given geometric configuration. This question is already fascinating in dimension . An example due to Keleti shows that there exists a subset of of Hausdorff dimension which does not contain any arithmetic progressions of length three. However, a result due to Laba and Pramanik ([9]) shows that there exists such that a subset of of Hausdorff dimension contains a progression of length three if it satisfies additional structural assumptions.
A similar difficulty arises in higher dimensions. An example due to Falconer ([4]) and (independently) Maga ([10]) shows that there exists a set of Hausdorff dimension in which does not contain vertices of an equilateral triangle. Once again, a result can be established with additional assumptions on the structure of the set and this was accomplished by Chung, Laba and Pramanik who proved the following, rather general result.
Definition 1.2.
Fix integers , , and . Suppose are matrices.
(a) We say that contains a point configuration if there exists vectors and such that
(b) Moreover, given any finite collection of subspaces with , we say that contains a nontrivial point configuration with respect to if and such that .
(c) Fix integers , , and . We say that a set of matrices is nondegenerate if
for any distinct indices .
Theorem 1.3.
[Chan, Łaba and Pramanik] Fix integers , , and . Let be a collection of nondegenerate matrices in the sense of part c) of Definition 1.2. Then for any constant , there exists a positive number with the following property: Suppose the set with supports a positive, finite, Radon measure with two conditions: (a) (ball condition) if , (b) (Fourier decay) Then (i) contains a point configuration in the sense of Definition 1.2 (a). (ii) Moreover, for any finite collection of subspaces with , contains a nontrivial point configuration with respect to in the sense of Definition 1.2 (b).
As the reader can check Theorem 1.3 recovers equilateral triangles in sets of Hausdorff dimension sufficiently close to under an additional structural assumption on sizes of balls and the decay rate of the Fourier transform. A natural question that arises at this point is what happens with triangles in dimensions and higher. The ChanLabaPramanik result does not address this issue, even under additional structural assumptions.
Before we state our main result, we would like to point out some interesting parallels between the configuration problems in Euclidean space and those in vector spaces over finite fields. Let denote the finite with elements and let denote the dimensional vector space over . Once again we may ask whether contains vertices of an equilateral triangle if is sufficiently large. In two dimensions there is a fundamental arithmetic obstruction, pointed out in [1], namely that if does not contain , then does not contain any equilateral triangles. It is not currently known whether a ”metric” rather an ”arithmetic” obstruction also exists. On the other hand, when , Hart and the first listed author proved in ([7]) that contains vertices of every possible congruence class of triangles, including equilateral triangles with every possible sidelength (in ). As the reader shall see in a moment, a very similar picture emerges in Euclidean space, even though the methods and technical formulations are very different.
1.1. Statement of results
The main result of this paper is the following. We recover equilateral triangles in sets of sufficiently large Hausdorff dimension without any additional assumptions on Fourier decay rate of the underlying measure.
Theorem 1.4.
Let be a compact subset of , and is a probablity Frostman measure on with for all . Then there exists such that if the , then contains vertices of an equilateral triangle.
1.2. Structure of the proof
Our proof consists of three basic steps:

STEP 1: To construct a natural measure on the set of equilateral triangles of a given sidelength with endpoints in and prove that it is finite.

STEP 2: To modify the ChangLabaPramanik argument to show that with the first step as input, the integral of the measure is strictly positive.

STEP 3: To show that this measure is not supported on the trivial ”singleton” equilateral triangles , thus guaranteeing the existence of a nontrivial equilateral triangle.
Notation. Throughout this paper, means there exists a constant such that .
2. Proof of Theorem 1.4
2.1. Proof of STEP 1
Denote by the surface measure of the surface
Define the measure on
by
(2.1) 
if the limit exists, where , and is supported in the unit ball with . The next result shows that the limit always makes sense for a sequence tending to zero if the Hausdorff dimension of the support of is sufficiently large.
Theorem 2.1.
(EXISTENCE) If , there exists a sequence such that the limit in (2.1) exists.
It suffices to show that
is bounded above by a constant independent in .
By Plancherel,
As a distribution,
where denotes the distribution at the origin, hence the integral above equals
(2.2)  
Since , we have
(2.3) 
The proof of (2.4) requires the following stationary phase estimate. The proof will be given later.
Lemma 2.2.
Suppose , then
(2.5) 
where is some rotation by and denotes the angle between and .
Assuming , then at least two of are .
Case 1: .
By Lemma 2.2,
(2.6)  
Fix ,
We shall need the following stationary phase argument that will be proven later on.
Lemma 2.3.
Lemma 2.3 implies that for each fixed with ,
Similarly, for each fixed with ,
Therefore, by Shur’s test, (2.6) is bounded by
(2.7)  
if . Case 2: . Pick any pair such that is an equilateral triangle. Notice
(2.8) 
where is the orthogonal group, is the probablity Haar measure on and it’s independent in the choice of . Thus
Since is an equilateral triangle implies that is also an equilateral triangle,
With
(2.9)  
by the argument in Case 1.
2.2. Step 2
In [3], the authors proved the following theorem
Theorem 2.4 (Proposation 5.1, [3]).
Let
where is nondegenerate (see Definition 1.2 above). Then for every , there exists a constant with the following property: for every function , , , we have .
It’s not hard to check that
where is the measure on the set of rotations induced by the map
Note that , . Fix , applying Theorem 2.4 with , , it follows that
Since the set of rotations is compact, after integrating in we have
The proof in section 2.1 implies
2.3. Step 3
The estimate (2.4) shows that if , as a function of ,
Dominated convergence theorem and the discussion in Step 2 implies that if , there exists and a sequence such that
is well defined and positive. This means there exists a positive measure
supported on
which completes the proof of Theorem 1.4.
3. Proof of Lemma 2.2
The upper bound of was first studied in [6] (Lemma 3.19), where the authors obtain
We follow the idea of the proof in the [6], with more delicate computation and observation. By partition of unity and the rotation invariance of (see (2.8)), one can only consider a small neighborhood of and . Introduce local coordinates on near , resp., , where we write
All calculations that follow will be modulo . Notice implies . However,
We use the quadratic terms in the implicit function theorem in one variable,
(3.1) 
to solve for in terms of , with as parameters:
Since is equivalent to , a neighborhood of can be parametrized by
modulo . Then the Fourier transform of the measure can be written as
and, by the invariance, we may asuume is a critical point, if exists.
The gradient of the phase function is,
And the Hessian is
Since is a critical point, if exists, it follows that
(3.2) 
Putting (3.2) and the first factor of the determinant of the Hessian together as a vector in ,
where is the vector obtained by rotating counterclockwise by .
This means, at the critical point,
where is some rotation by .
4. Proof of Lemma 2.3
If and , the lemma is trivial. If is small,
Using polar coordinate we see that
where the last equality comes from the formula . If is small, it follows that since . Thus with