On GILP's group theoretic approach to Falconer's distance problem

In this paper, we follow and extend a group-theoretic method introduced by Greenleaf-Iosevich-Liu-Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP's theorem in [GILP15]. This allows us to extend the Wolff-Erdogan dimension bound for distance sets to finite points configurations with $k$ points for $k\in\{2,\dots,n+1\}.$ At the end of this paper, we extend this group-theoretic method and illustrate a `Fourier free' approach to Falconer's distance set problem for the Lebesgue measure. We explain how to use tubular incidence estimates in distance set problems. Curiously, tubular incidence estimates are also related to the Kakeya problem.


HAN YU
Unlike most of the results which follow from harmonic analytic methods, this 3s/2 bound holds for s < 1 as well. For distance sets (k = 2, n ≥ 2), one can obtain a similar result which says that the upper box dimension of D(F) is at least s/n. We note here that the s/n bound is often strict, see [6].
Following the approach in [7], we make the following definition. DEFINITION 1.2. Let F ⊂ R n be a compact set and let μ be a probability measure supported on F. For g ∈ O(n), the orthogonal group on R n , we construct a measure ν g as follows, here C 0 (R n ) is the space of continuous functions with compact support on R n . In other words, ν g = μ * gμ. We also construct a measure ν on k (F) ⊂ R k(k−1)/2 by where t is a k(k − 1)/2-vector with entries |x i − x j | for 1 ≤ i < j ≤ k.
In this way, we see that ν is 'the natural measure' supported on k (F). In particular, we have dim H ν ≤ dim H k (F). We will introduce some notions of dimensions in the following section. Notice that our definitions are slightly different than those in [7]. Here, we use k to denote the number of vertices of the 'simplex structures' we want to count in F while in [7], k is the order of the simplices. For example, when k = 2, our definition gives distance sets while the definitions in [7] gives triangle sets.
The above result generalises [7,Theorem 1.3] in two ways. First, it provides us a good estimate of the growth of ν δ 2 2 with respect to δ → 0, which in turn allows us to estimate FALCONER'S DISTANCE PROBLEM 549 the Hausdorff dimension of k (F). Here, ν δ represents a δ-scale smooth approximation of the measure ν. More precisely, it is ν * φ δ where ψ δ = δ −k(k−1)/2 φ(./δ) for a smooth cut-off function φ on R k(k−1)/2 . See Section 2 for more details. Second, we can drop the technical continuity condition mentioned above. In this way, the above theorem can be seen as an alternative approach to the dimension results of distance sets discussed in [13,Chapter 15]. We record the Hausdorff dimension estimate as a corollary.
where γ s is the same quantity as in the statement of Theorem 1.3.
We will prove the above result in Section 4.2. For example, when k = 3, n = 2 we have If 2.5s − 1 > 3, that is, s > 8/5, then 3 (F) would have positive Lebesgue measure. This is a result proved in [7].
We have another consequence from Theorem 1.3. We can cover R n with closed δ-cubes K δ with disjoint interiors. For each K ∈ K δ , we use 2K to denote the 2δ-cube with the same centre as K. Observe that Since {2K} K∈K δ covers R n with maximal multiplicity 2 n+1 , we see that By Theorem 1.3 and the argument above, we see From here, we deduce the following corollary.
COROLLARY 1.5. Let μ be a s-Frostman measure with compact support on R n . Let k ∈ {2, . . . , n + 1} be an integer and ν g , ν be as in Definition 1.2. If −((n − s)(k − 1) − γ s ) > 0, then for almost all g ∈ O(n), ν g is an L k (R n ) function. In particular, for such g ∈ O(n), ν g is absolutely continuous with respect to the Lebesgue measure. Here γ s can be chosen as in Theorem 1.3.

HAN YU
If s = n, then ν g is an L ∞ -function for almost all g ∈ O(n). For k = 2, we see that the positivity criterion happens when This recovers a result stated at the end of [13,Section 15.5]. In Section 7, we give some sketched discussions in this situation. In [13,Section 7.3], it was asked whether the following conjecture is true. CONJECTURE 1.6. Let μ be a s-Frostman measure with compact support on R n . If s > n/2 then for almost all g ∈ O(n), ν g is absolutely continuous with respect to the Lebesgue measure. Now there are better results than (1.1), see [9] and the references therein for more details.

Notation.
1. Let f be a function on R n , we writef for its Fourier transform, where ω ∈ R n and (ω, x) is the Euclidean inner product between ω and x. Let μ be a probability measure on R n we also writeμ for its Fourier transform, 2. For each integer n ≥ 1, we will often need to find a smooth cut-off function φ n on R n . More precisely, we define φ to be 1 on the unit ball and 0 outside the ball of radius 2 centred at the origin. Then, we can smoothly construct this function φ n . Let δ > 0 we write φ δ,n to be the function Throughout this paper, when the ambient space of ψ n is clear, we will just write it as φ. Let f be a function on R n we write f δ = f * φ δ . Similarly for a measure μ, we write μ δ = μ * φ δ .
3. It is convenient to introduce notions ≈, < ∼ , > ∼ for approximately equal, approximately smaller and approximately larger. As our estimates always involve scales, we use 1 > δ > 0 to denote a particular scale. Then for two quantities f (δ), g(δ), we define the following: We will use the same symbols for scales tending to ∞ as well. More precisely, for R ∈ (0, ∞), and quantities f (R), g(R), we write if there is a constant C > 0 such that f (R) ≤ Cg(R) for all R > 0. Similar meanings can be given to symbols > ∼ and ≈ .

Hausdorff dimension for sets.
Let n ≥ 1 be an integer. Let F ⊂ R n be a Borel set. For any s ∈ R + and δ > 0, define the following quantity More details about the Hausdorff dimension can be found in [5] and [12].

Frostman's measure.
It is known (e.g., see [13,Theorme 2.7]) that if F is a Borel subset of R n with dim H F = s, then for any > 0 there is a measure μ supported in F such that for all x ∈ F and r > 0 we have μ(B(x, r)) ≤ r s− . Such a measure μ is usually called a (s − )-Frostman measure.

Energy integrals and Hausdorff dimension for measures.
Let μ ∈ P(R n ), the space of Borel probability measures on R n . For each positive number t > 0, we define the t-energy of μ to be Through Fourier transform, it can be shown that where γ (n, s) = π s−n/2 ((n − s)/2)/ (s/2) and when s ∈ (0, n) we have γ (n, s) ∈ (0, ∞), see [13, Sections 3.4 and 3.5]. We define the Hausdorff dimension of μ as follows: Let F ⊂ R n be a Borel set, then we have This implies that if μ ∈ P(F), we have dim H μ ≤ dim H F.

Spherical averages and Wolff-Erdogan's estimate.
Let μ ∈ P(R n ). We define the following spherical average forμ, where dσ is the normalised Lebesgue measure on S n−1 . We have the following deep result on the decay rate of S(μ, R) as R → ∞, see [4,17]. The following version is taken from [13,Theorem 15.7]. 552 HAN YU THEOREM 3.1 (Wolff-Erdogan estimate). Let μ ∈ P(R n ) with compact support, for each s ≥ n/2, > 0, there is a positive constant C(n, s, ) and for all R > 0, we have Here γ s can be chosen as follows: 3.6. Group-theoretic energy. Let n ≥ 2 be an integer and k ∈ {2, . . . , n + 1}. Let μ ∈ P(R n ). For each δ > 0, g ∈ O(n), we define k-group-theoretic energy for μ at scale δ with respect to g to be Often, we can write E(μ, g, δ) for E k (μ, g, δ) as the dependence on k will be always assumed. If A ⊂ R n is a finite set and μ is the normalised counting measure on A. Let k = 2, δ = 0, we see that which counts the number of quadruples (x 1 , This idea was introduced in [3] and it played a crucial role in Guth-Katz's proof of Erdős' distance problem, see [10] and [8, Section 9].
4. An L 2 approach to the Hausdorff dimension.

Some general results.
In this section, we discuss a simple method for estimating the Hausdorff dimension of a compactly supported Borel probability measure μ in P(R n ). We denote its Fourier transform asμ. It is a continuous function as μ is compactly supported. In general it is not L 2 , for otherwise μ is in fact an L 2 function. To measure how far away it is from being L 2 , we take the following ball average, see also [13,Section 3.8], If lim R→∞ A(μ, R) < ∞, then μ can be viewed as an L 2 function. In general, we expect that A(μ, R) tends to ∞ at a certain speed. If there is a constant C > 0 and a number s > 0 such that for all R > 0, then we see that for t ∈ (0, n) Since μ is a probability measure,μ is bounded on unit ball. Therefore, we see that |ω|≤1 |μ(ω)| 2 |ω| t−n dω < ∞.
For each j ≥ 0, we have If s + t − n < 0, the sum with respect to j converges and we have Therefore, dim H μ ≥ t whenever t < n − s. This implies that dim H μ ≥ n − s. (4.1) In order to study this L 2 phenomena more systematically, we introduce the following notion of dimension, There are several other ways of doing this L 2 approach. For example, we can define if u − t > 0 or else the above sum is bounded uniformly for all R. In terms of the L 2dimension, we see that if dim L 2 μ > n − t then In general, A(μ, R, |.| −t ) could have a smaller growth exponent. It is interesting to find the infimum among all possible values s such that holds for all R > 0. More precisely, we consider the following quantity In general, it is possible that the above inequality is strict. By collecting the results (4.1) and (4.2), we have shown the following result.
THEOREM 4.1. Let n ≥ 1 be an integer and μ ∈ P(R n ) be a Borel probaility measure. Then, we have The function t ≥ 0 → dim L 2 ,t μ is non-increasing and bounded from above by dim L 2 μ.
In most cases, it is difficult to estimate A(μ, R) directly. A useful method is to consider the L 2 -norm of μ δ = μ * φ δ . Notice that μ δ is a Schwartz function taking non-negative values. Sinceμ δ =μφ δ andφ δ decays very fast outside the ball B δ −1 (0), we see that where the implicit constant in < ∼ depends only on the choice of the cut-off function φ.

Wolff-Erdogan bound for finite points configurations: Proof of Corollary 1.4.
Before we prove Theorem 1.3, let us see how to obtain a Hausdorff dimension estimate. Let F ⊂ R n and dim H F = s. Then, we can choose (s − )-Frostman measure on F for each > 0. Then by Theorem 1.3 together with the discussions above, we see that provided that the RHS is not greater than k(k − 1)/2, otherwise, k (F) has positive Lebesgue measure. For k = 2, this result revisits the Wolff-Erdogan-Mattila's bound for the Hausdorff dimension of distance set.

Proof.
A proof can be found in [7, Section 2].
LEMMA 5.2. Let n ≥ 2 and k ∈ {2, . . . , n + 1} be integers. Let μ ∈ P[0, 1] n and ν g , ν as defined before. Then for each δ > 0, g ∈ O(n), we have Proof. By putting in definitions, we see that the statement of this lemma is equivalent to μ 2k {(x 1 , . . . , x k , y 1 , . . . , y k To prove this, let x 1 , . . . , x k−1 , y 1 , . . . , y k−1 be fixed, consider the following section It is easy to see that the above section is contained in We see that the μ 2k measure is now bounded from above by . By the definition of ν g , we see that gy 1 )).
If ν g does not give positive measure on any spheres, then we would get gy 1 )).

HAN YU
However, we do not assume this continuity of ν g and we only have an upper bound. We can do the above step k − 1 times and by Fubini's theorem we see that If ν g (B 2δ (.)) would be continuous, then we would have In general, we choose a continuous function sandwiched by ν g (B 2δ (.)) and ν g (B 2.5δ (.)) (by taking convolution with a suitable smooth cut-off function), and then apply the definition of ν g to arrive at the above inequality.

The main result.
In this section, we give a detailed proof of Theorem 1.3. We note that in [7], a proof is given under the condition that ν g is absolutely continuous for almost all g ∈ O(n). In [18], a sketched proof is given for the case when k = 3 and we note that the same strategy works for general cases k ≥ 2 as well and here we will provide more details.
Proof of Theorem 1.3. By Lemmas 5.1 and 5.2, we see that as δ → 0, where C > 0 is a constant. The situation would be simple if ν g (B 2.5δ (z)) would be continuous with respect to z. However, we cannot assume this continuity condition. To deal with this issue, let φ DD (.) be a radial Schwartz function such thatφ DD is real-valued, nonnegative, vanishes outside the ball of radius 0.5c > 0 around the origin and is equal to a positive number c > 0 on a ball of radius c > 0 around the origin. Now we take the square φ D = (φ DD ) 2 and see thatφ We see thatφ D is radial, real-valued, non-negative, vanishes outside the ball of radius c around the origin. Unlikeφ DD ,φ D is no longer a constant function on any ball centred at the origin. By further rescaling if necessary, we may assume that φ D (x) ≥ 1 for x ∈ B 2.5 (0). This can be done because φ D is real-valued, Schwartz and φ D (0) > 0. Sinceφ D is compactly supported, we can denote c = φ D ∞ . Then we write h g,δ = ν g * φ D (δ −1 .). We see that Now we write f g,δ (.) = δ −n h g,δ (.), as a result we see that Let ψ be a smooth cut-off function supported in {ω ∈ R n : |ω| ∈ [0.5, 4]} and identically equal to 1 in {ω ∈ R n : |ω| ∈ [1, 2]}. We can also require that j∈Z ψ(2 −j ω) = 1 and this is the starting point of the Littlewood-Paley decomposition. Let f g,δ,j , ν g,j be the j-th Littlewood-Paley piece of f g,δ , ν g , respectively, namely,f g,δ,j (ω) =f g,δ (ω)ψ(2 −j ω) and similarly for ν g,j . We need to bound f g,δ,j ∞ as well as ν g,j ∞ . The later can be bounded by C 2 j(n−s) for any s < dim H F with a constant C depending on the function ψ. This was shown in [7, p. 805]. For the former, we will be interested in estimating f g,δ,j ∞ when 2 j is not as large as δ −1 . In this case, recall that f g,δ = ν g * φ D δ and in terms of Fourier transform we havef Then we see that By the discussion in [13,Section 3.8], we see that The same estimate holds for B 2 j+2 (0) |μ(gω)| 2 dω as well. Therefore, we see that where C > 0 is a constant which does not depend on g, j, δ. Observe that if 2 j−1 > c δ −1 , then f g,δ,j = 0 and this is the reason for considering 2 j to be not much larger than δ −1 . Thus, we have obtained a complete estimate for f g,δ,j ∞ .
In what follows, we want to estimate the following integral: We want to apply the argument in [7, Section 3] and we provide details depending on whether k = 2 or k ≥ 3. We note here that the argument in [7, Section 3] works only for k ≥ 3 but we shall extend it to the case when k = 2.

HAN YU
We note here thatf g,δ (ω) =f g,δ (−ω). Therefore, we see that Recall that f g,δ = ν g * φ D δ , we see thatf Then since ν g = μ * gμ, we see that As a result, we see that Observe thatφ D δ is a cut-off function at scale δ −1 . More precisely, for |ω| > c δ −1 , we havê By integrating first with respect to dg and then dω, we see that where dσ is the Lebesgue probability measure on S n−1 . We writeφ D δ (t) =φ D δ (ω) for |ω| = t and similarly for ψ(2 −j t). Since ψ and φ D δ are radial functions, the above step is well defined. The constant C(n) is a positive number which depends only on n.
We need to sum (6.2) over j ∈ Z. Because of the cut-off property of φ D δ , we only need to consider the sum up to j:2 j ≤2c δ −1 . More precisely, there is a positive constant C > 0 and we have In fact when 2 j > 2c δ −1 , then ψ(2 −j ω)φ D δ (ω) = 0. Therefore, we do not need to sum larger values of j. This is because we can choose a special cut-off function φ D whose Fourier transform is compactly supported. This makes φ D not compactly supported but we do not need this. We still need to sum negative values of j but as μ is a probability measure we have where the above inequality holds for all > 0. As in the case when k = 2, we see that, f k−1 g,δ (z)dν g (z)dg < ∼ C(k, s, ν) + j:1≤2 j ≤2c δ −1 2 −j(γ s − ) 2 j(n−s) 2 j(n−s)(k−2) .
If (n − s)(k − 1) − γ s + < 0, then ν would be an L 2 function, otherwise we see that This concludes the proof for the case when k ≥ 3.

Asymmetric distance sets.
Let n ≥ 2 be an integer. Let F 1 , F 2 are compact sets in R n with dim H F 1 = s 1 , dim H F 2 = s 2 . Let μ 1 , μ 2 be probability measures supported on F 1 , F 2 , respectively. For g ∈ O(n), the orthogonal group on R n , we construct a measure ν g as follows: In other words, ν g = μ 1 * gμ 2 . We also construct a measure ν by It can be seen that ν is supported on Most of the argument in previous sections can be used here. In particular, one can show that for each > 0 Therefore, we see that if max{γ s 1 + s 2 , γ s 2 + s 1 } > n, then D(F 1 , F 2 ) has positive Lebesgue measure. If s 2 ≥ s 1 , then this is equivalent to s 2 + 0.5s 1 > 0.75n + 0.5. Now we turn to Corollary 1.5. With the same arguments as above, we see that if s 2 + 0.5s 1 > 0.75n + 0.5, then for almost all g ∈ O(n), ν g is absolutely continuous with respect to the Lebesgue measure. In general, we can consider k ≥ 3 and obtain conditions for ν g to be L k for almost all g ∈ O(n).