On restricted Falconer distance sets

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, $k$-point configuration sets given by $$\Delta^{diag}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}$$ for a compact $E\subset\mathbb{R}^d$ and $k\ge 3$. We show that $\Delta^{diag}(E)$ has non-empty interior if the Hausdorff dimension of $E$ satisfies \begin{equation*} \dim(E)>\begin{cases} \frac{2d+1}3,&k=3 \\ \frac{(k-1)d}k,&k\ge 4. \end{cases} \end{equation*} We prove an extension of this to $C^\omega$ Riemannian metrics $g$ close to the product of Euclidean metrics. For product metrics this follows from known results on pinned distance sets, but to obtain a result for general perturbations $g$ we present a sequence of proofs of partial results, leading up to the proof of the full result, which is based on estimates for multilinear Fourier integral operators.


The Falconer distance problem and its many variants
The Falconer distance problem, a continuous analogue of the celebrated Erdős distance problem asks: How large does dim(E), for a compact set E ⊆ R d , need to be to ensure that the Lebesgue measure of its distance set ∆(E) = {|x − y|, x, y ∈ E} is positive?Here and below dim(E) denotes the Hausdorff dimension of the set E. Falconer introduced this problem in 1985 in [10] and established the dimensional threshold dim(E) > d+1 2 .Further, Falconer conjectured the threshold is dim(E) > d 2 and showed the result could not hold true strictly below that threshold.Falconer's problem has stimulated much activity and been the focus of many outstanding results, e.g., [5,31,9,6,8,18,7].
For two compact sets E, F ⊆ R d one can also consider an asymmetric version, given by ∆(E, F ) := {|x − y| : x ∈ E, y ∈ F }, so that ∆(E, E) = ∆(E).Note all the standard proofs adapt to this setting and the threshold condition can be replaced by a lower bound on (dim(E) + dim(F ))/2.
Yet another variant of the Falconer problem was introduced by Mattila and Sjölin [26], who asked how large does dim(E) need to be in order to ensure that ∆(E) satisfies the stronger condition of having nonempty interior, and showed that dim(E) > d+1 2 is sufficient.Both the Falconer and Mattila-Sjölin problems have pinned versions, asking how large does dim(E) need to be to guarantee that there exists an x such that the pinned distance set, ∆ x (E) := {|x − y| : y ∈ E}, has positive Lebesgue measure or nonempty interior.Peres and Schlag [30] showed that this holds for dim(E) > d+2 2 , d ≥ 3; see [19] for some improvements and generalization.More recently, improvements to thresholds in the Falconer's distance problem automatically transfer over to the pinned setting due to the magical formula of Liu [23].
Nowadays one can view the original result of Falconer as well as this one of Mattila and Sjölin through the same lens; see, e.g., [25].As with Falconer's original problem, this has led to considerable further work in more general settings [20,15,16,27,22,28,17].

A new problem and motivation
In this paper we introduce new variants of the Falconer and Mattila-Sjölin problems, which we call restricted distance problems. 1These lie between the original distance problems and and their pinned variants, and when stated in general encapsulate both of them.
For a compact set E ⊆ R d , let F ⊆ R d be a compact set which might depend on E. Defining the restricted distance set, we ask what lower bounds on dim(E) guarantee that ∆ F (E) has positive Lebesgue measure or nonempty interior.Note that if F has no dependence on E then ∆ F (E) = ∆(E, F ) and so we are in the asymmetric setting of the Falconer distance problem.The simplest case of a set F which is dependent on E is when F = E, so that ∆ F (E) = ∆(E). 1After the original version of this preprint was posted, Borges, Iosevich and Ou posted [4], which also discusses restricted distance problems and in some cases obtains lower thresholds than we obtain here.See § §3.1 for a discussion.However, we believe that Thm.3.3 is not currently accessible to the methods of [4], and in any case the techniques used to proved it indicate that positive results for restricted Mattila-Sjölin type problems can be proven in great generality.
If F = {x 0 } for some point x 0 , fixed in advance, then this is similar to a pinned distance problem, but stronger than the usual one, since the pin is fixed.(A result giving nonempty interior for the set of volumes of parallelepipeds generated by an arbitrary x 0 and all d-tuples of points in E ⊂ R d is in [16,Thm.1.2],where this is referred to as a strongly pinned result.) To illustrate the type of Falconer-type distance problems in which we are interested, we focus on a prototype lying between the original and pinned versions of the distance problem.For a compact set which we will also denote by ∆ diag (E), has positive Lebesgue measure or nonempty interior.See Fig. 1 below.
As noted in [4], in order to make the problem more interesting, in (2.1) one should impose a condition y 1 ̸ = y 2 because, if y 1 = y 2 were allowed, then ∆ diag (E) ⊃ √ 2 • ∆(E), which would then have positive Lebesgue measure or nonempty interior if dim(E) is greater than the thresholds in R d for the standard Falconer or Mattila-Sjölin distance problems, resp.We thus include this condition and its extensions in the statements below.So, we define for an l-fold Cartesian product of a E ⊂ R d , and an where | • | is the Euclidean norm on R ld .
We are now ready to pose the following set of questions generalizing the prototype: Restricted Falconer and Mattila-Sjölin Problems.Fix l ∈ N and a map F from the collection C R d of compact sets in R d to 2 C(R ld ) , denoting the image of a compact E by F E .
Q. What lower bounds on dim(E) ensure that either (i) there exists an F ∈ F E such that ∆F (E) has positive Lebesgue measure (or nonempty interior); or (ii) for a.e.F ∈ F E (with respect to some measure on F), or (iii) for every F ∈ F E , the same property holds.Remarks. 1.For l = 1, case (i) and positive Lebesgue measure, the choice of F E = {E} yields the classical Falconer distance problem, while F E = {{x} : x ∈ E} yields its standard pinned variant.On the other hand, ∆diag (E) corresponds to l = 2, F E = {F }, where F is the diagonal of E × E in R 2d .In general, if F is a singleton, the questions (i), (ii) or (iii) collapse into one, concerning a 3-point configuration problem of either Falconer or Mattila-Sjölin type; in this paper we will focus on the latter for ∆diag (E) and its k-point configuration generalizations.
2. R d can be replaced by a smooth d-dimensional manifold with a smooth density, and Thm.3.3 below is formulated in this setting.

3.
Returning to the prototype (2.1), note that if we don't restrict to the diagonal but instead consider the full R 2d distance set ∆(E × E), the best results known for the R 2d Falconer problem would yield a sufficient lower bound on dim(E).Since dim(E) , and 2d is even, the results of [18,7] yield that ∆(E × E) has positive Lebesgue measure.However, for the restricted Falconer problem we are considering, the set ∆diag (E) consists only of distances from points on the diagonal of E to general points of E × E.

4.
By a result of Peres and Schlag [30], if dim(E) > (d + 2)/2, with d ≥ 3, then there exists an x such that the pinned distance set ∆ x (E) = { |x − y 1 | : y 1 ∈ E } contains an interval.This immediately implies that ∆diag (E) contains an interval, since y 2 in (2.1) can simply be fixed.The same principle applies to any ∆F (E) with F of the form with arbitrary continuous functions ϕ j : R d → R d .Further comments are in § §3.1.below.However, this argument relies on both the form of F and the product nature of the Euclidean metric on R ld , and thus does not apply to our most general result, Thm. 3.3.

The main results
Our main results are the following, in increasing order of generality.
This result is the 3-point configuration set case of More generally, we have For any g ∈ G, let d g be the induced distance function, which is defined on at least a neighborhood of the diagonal of R d k−1 .Let g 0 denote the Euclidean metric.Then there is an N = N d,k ∈ N and a neighborhood U of g 0 in the C N topology on G such that if g ∈ U, and for a compact E ⊂ R d we define 3.1.Relations with known results.As explained in Remark 4 above, a result for the pinned Mattila-Sjölin problem in R d automatically yields nonempty interior for ∆F (E) whenever F is of the form (2.2), which includes the (k − 1)-fold diagonal.Thus, the pinned distance set threshold of dim(E) > (d + 2)/2, d ≥ 3, from Peres-Schlag [30] produces a better result than Thm.3.1 for d ≥ 4, and similarly for [19] for d ≥ 5. however, Thm.3.1 is better for d = 3, and for d = 2, where [30] doesn't apply.Similarly, [30] yields for k ≥ 4 a threshold at least as good as Thm.3.2 in all d ≥ 3.
The very recent paper of Borges, Iosevich and Ou [4] gives a lower threshold than our Thm.3.1 in all dimensions.The authors state that their method extends to the context of Thm.3.2, but without giving specific thresholds.On the other hand, it is not clear that the technique of [4] applies in the setting of Thm.3.3, due to the non-product nature of general Riemannian metrics g on It is reasonable to ask why we are persisting in stating and proving Thms.3.1 and 3.2.The point is that, rather than proving Thm.3.3 immediately, we will build up to it with a proof of Thm.3.1 based on an L 2 × L 2 → L 2 decay bound for a bilinear spherical averaging operator.This naturally leads to the multilinear operators and estimates yielding Thm. 3.2, which we analyze and prove in the Fourier integral operator framework of Greenleaf, Iosevich and Taylor [16].With minimal additional effort, this then leads to Thm. 3.3 in the case of a product metric; the inherent stability of the FIO approach under general perturbations then allows it to be proven in full generality.
We now start with the proof of Thm.3.1.

The Bilinear Spherical Averaging operator
Let d ≥ 2. Then for x ∈ R d , t > 0 and for functions f, g ∈ S(R d ) we define the averaging operator: where σ is the surface measure on unit sphere S 2d−1 in R 2d , Next, we define the (full) maximal version of the bilinear spherical operator Finally, we define its single-scale (localized) bilinear maximal operator, which is Known results and goals.The operators A r and M first appeared in the paper of Geba, et al., [12], where the authors proved some initial L p improving estimates for these operators.Subsequently, the L p improving estimates for M were further developed in the works of Barrionevo-Grafakos-D.He-Honzík-Oliveira (see [1]), Grafakos-D.He-Honzík (see [14]) and Heo-Hong-Yang (see [11]).Finally, the full region L p improving estimates for the operator M were given in the work of Jeong and Lee (see [21]) as the result of a clever "slicing" argument enabled them to pointwisely dominate the maximal biliner spherical averaging operator by the product of a Hardy-Littlewood maximal operator and a linear spherical averaging operator, both of which have been extensively studied.Furthermore, in the same work the authors explored the L p improving estimates for the operator M getting a large region of exponents, however there is still work left open in this case.Subsequent developments have included sparse domination results [29,3] and very recent lacunary maximal operator results [2].
For our work, we need not only L p improving estimates but more specifically L 2 × L 2 → L 2 estimate with decay.We already know there operator is bounded from L 2 × L 2 → L 2 but this is not enough.If one considers functions which have compact support on the frequency side, then a decay factor appears.Moreover, we just consider the operator A r and given the absence of supremum in the definition, one could exploit the full decay of the surface measure on the unit ball in R 2d .
We start with the following proposition.Proposition 4.1.Let i, j ∈ N and let f, g be functions with 5. Proof of Theorem 3.1 In this section we will give a proof of Thm.3.1 using the bilinear spherical averaging operator.We start by proving Proposition 4.1.
5.1.The decay of the measure σ.We define σ r to be the surface measure on the sphere of radius r in R d × R d : Then by the classical method of stationary phase we have for (ξ, By a change of variable we get Next, for ξ ∈ R d we use the Fourier inversion formula and Fubini's theorem to write r 2d−1 A r (f, g) (ξ) as: Proof.We can assume without loss of generality that i ≤ j.Then, we apply Plancherel's theorem to get: Using the decay of the measure in (5.1) we get Note that the inner integral is supported on this set and so we can estimate it by using Cauchy-Schwarz inequality: which gives, after applying Fubini's theorem and a change of variable: This finishes the proof of Proposition 4.1.□

Conclusion of proof of Theorem 3.1.
Proof.Let E ⊂ R d with dim(E) > 2d+1 3 .Then, fix s ∈ ( 2d+1 3 , dim(E)).We argue as in the proof of Theorem 4.6 in [25].By the Frostman's Lemma [25, Thm.2.8], there exists a measure µ ∈ M(E) with I s (µ) < ∞.Then we define its distance measure δ(µ) ∈ M( ∆diag (E)) defined for Borel sets B ⊂ R d by where we seemingly have enlarged the integrand by adding in the case when y 1 = y 2 , but note that µ × µ((y 1 , y 2 ) : y 1 = y 2 ) = 0 which follows from the Frostman condition because the set is of strictly lower dimension than E ×E.In other words, δ(µ) is the image of µ×µ under the distance map (x, y 1 , y 2 ) → |(x, x) − (y 1 , y 2 )|, or equivalently for any continuous function g on R: Next, for a smooth function f with compact support we have δ(f ) is also a function.To see this we write: where in the third equality for y In the third equality we used polar coordinates where ω = (ω 1 , ω 2 ) ∈ S 2d−1 .Therefore, after applying Fubini's theorem and using the definition of the Bilinear Averaging Spherical operator A r we have Next, we will approximate weakly the measure µ mentioned above.Namely, let ψ be a smooth compactly supported function in R d with ψ = 1.As usually, we define ψ ϵ (x) = ϵ −d ψ( x ϵ ) and µ ϵ = ψ ϵ * µ.Then µ ϵ → µ, as ϵ → 0 weakly and so δ(µ ϵ ) → δ(µ) weakly.Moreover As we know, for any ϵ > 0, µ ϵ , is a function.Then, from the formula for δ(f ) we get and by the comments above the left side converges weakly to δ(µ)(r).We would like to see where the right hand side converges to.Using Parseval's theorem we see Next, we have lim ϵ→0 µ ϵ (ξ) = µ(ξ) pointwise, and since with passing the limit inside justified by the Dominated Convergence Theorem.Note that and the last function is integrable (in η) by utilising the Cauchy-Schwarz inequality, a change of variable and the fact I1 2 (µ) ≤ I s (µ), since µ has compact support and s > 1  2 .Now we write We will dominate each of these functions by L 2 integrable functions, independently of ϵ and so B ϵ (ξ) E ϵ (ξ) will be dominated by an L 1 function, independently of ϵ which will allow us to use the dominated convergence theorem and get the formula and the L 2 norm of the right side is exactly equal to I s (µ) which is finite. Secondly, Now we will decompose μ on dyadic scales.Consider the Schwartz functions η 0 (ξ) supported at |ξ| ≤ 1  2 and η(ξ) supported in the spherical shell 1  2 < |ξ| ≤ 2 such that the quantities η 0 (ξ), η j (ξ) := η 2 −j ξ , with j ≥ 1, form a partition of unity.
Then we define µ j (x) := µ * η j (x) and so µ j (ξ) = µ(ξ)η j (ξ).Thus, on the supports of µ i , µ i and d > s.Now the function on the right is independent of ϵ and L 2 integrable as by Minkowski's integral inequality and Proposition 4.1.Next, we want to evaluate the L 2 norms of the functions µ i .We have by Plancherel's theorem: With this at hand, we continue estimating I by utilizing the symmetry of the summand which is finite since s > 2d+1 3 .Therefore, for a set E with dim(E) > 2d+1 3 we have that the function in (5.2) is continuous, as can be seen by the Dominated Convergence Theorem.Next, since supp(δ(µ)) ⊂ ∆ diag (supp(µ)) ⊂ ∆ diag (E) it follows that ∆diag (E) has non-empty interior.□ Remark 5.1.The same proof works for an arbitrary number of points.
extending what we have just shown for k = 3.However, it turns out that by using the Fourier integral operator (FIO) approach of [15,16], for k ≥ 4 one can lower this by 1/k.More importantly, the FIO approach does not require the metric to be Euclidean, or a product, or even translation invariant, leading to Thm. 3.3..

Thm. 3.2 by a Fourier integral operator approach
We now prove Theorem 3.2 using multilinear Fourier integral operators (FIO), improving on the threshold in Remark 5.1 for k ≥ 4. Using the FIO method developed in [15] for 2-point configuration sets and then extended in [16] to k-point configurations by optimizing linear FIO estimates over all bipartite partitions of the variables, this will then set the scene for the proof of Thm.3.3.
We will give the calculations needed to prove Theorem 3.2, using the general framework and notation of [16], which the reader should consult for a full exposition.In the terminology of [13], the k-configuration set ∆diag k (E) is a Φ-configuration set.For convenience, we relabel (x, y 1 , . . ., y k−1 ) as (x 0 , x 1 , . . ., x k−1 ) and define Φ : has nonempty interior.
We start by finding a base point in R kd about which to work.Let s 0 = s 0 (d, k) be the threshold for dim(E) in (3.1) in the statement of Thm.3.2, and suppose dim(E) > s 0 .Pick an s with s 0 < s < dim(E), let µ be a Frostman measure supported on E and of finite s-energy (see [24,Thm. 8.17]).We claim that there exist points x 0 0 , . . ., x k−1 0 ∈ E and a δ > 0 such that µ(B(x i 0 , δ)) > 0, 0 ≤ i ≤ k − 1, x i 0 ̸ = x 0 0 , ∀ i > 0, and To see this one can argue as in [16, § §4.1].The key point is that if we define (6.3) W := (x 0 , . . ., x k−1 ) ∈ R kd : x i ̸ = x 0 , ∀ i > 0, and then W is a Zariski open subset of R kd , whose complement is contained in a union of algebraic varieties of dimensions ≤ (k − 1)d (since each [16], where this type of argument is given for several different Φ-configurations, for more details.
For each t > 0, the configuration function Φ induces a surface measure, supported on the incidence relation (6.4) where N * Z t ⊂ T * R kd \ 0 is the conormal bundle of Z t .For convenience, we will write N * Z t with each pair of spatial and cotangent variables, (x i , ξ i ), grouped together.Thus, To make this more explicit, we parametrize an open subset of Z t by letting x 0 range freely over R d , and then write and let Ůt := (⃗ y, y k−1 ) ∈ R (k−1)d : and which is an open subset of R (k−1)d .Since all of the x i − x 0 = y i are distinct, it follows that y i + r(⃗ y, t)ω ; x 0 + y 1 , τ y 1 ; . . .; x 0 + y k−2 , τ y k−2 ; x 0 + r(⃗ y, t)ω, τ r(⃗ y, t)ω (6.6) Note that (x 0 0 , x 1 0 , . . ., x k−1 0 ) ∈ Zt 0 , with t = t 0 as in (6.2).Multiplying K t by a smooth cutoff function in order to localize to where Zt , which for simplicity we still denote by K t .
With all of this in place, we commence the proof of Thm.3.2 by treating the case k = 3, showing how the FIO approach reproves Thm.3.1.Since the number of points is odd, we need to consider partitions of the form σ = ( σ L | σ R ) = ( i | j k ), with i, j, k ∈ {0, 1, 2} distinct, and in fact we focus on σ 0 := ( 0 | 12 ).This corresponds treating K t 0 as the Schwartz kernel of a linear FIO, T σ 0 t 0 , taking functions of x 1 , x 2 to functions of x 0 .From (6.6), the canonical relation C σ 0 t for general T σ 0 t simplifies to x 0 , −τ y 1 + r(y 1 , t)ω ; x 0 + y 1 , τ y 1 ; x 0 + r(y 1 , t)ω, τ r(y 1 , t)ω is injective.This uses the fact that the radial derivative of r(⃗ y, t) with with respect to y k 2 is nonzero.(We could have used y i • ∂ y i for any of the variables y finishing the proof of Thm.3.2 for k ≥ 4 and even. Finally, for k ≥ 5 and odd, parity prevents the existence of an equidimensional partition, as it did for k = 3. Choosing Hörmander's estimates for FIO on R d or a d-dimensional smooth manifold are stable with respect to perturbations (in C ω ) which are small in the C N topology on the canonical relations and amplitude, for some N = N d .(This might not be stated explicitly in the literature, but is a folk theorem, being clear from the proofs.)Due to this stability, Thm.3.3 follows almost immediately from the proof of Thm.3.2 above.Perturbing the Euclidean metric g 0 on R (k−1)d in the C N +3 topology (for N = N (k−1)d ) results in a C N +1 perturbation of the geodesic flow, and hence a C N +1 perturbation of the distance function.Thus, the configuration function, Φ g (x 0 , . . ., x k−1 ) = 1 2 d g (x 0 , . . ., x 0 ) , (x 1 , . . ., x k−1 ) 2 is a C N +1 perturbation of Φ g 0 (x 0 , . . ., x k−1 ) = 1 2 (x 0 , . . ., x 0 ) − (x 1 , . . ., x k−1 ) 2 , which was the starting point (6.1) for the analysis in the previous section.The existence of a base point (x 0 0 , x 1 0 , . . ., x k−1 0 ) ∈ R kd around which to run the whole argument follows as before, since the analogue of W in the Riemannian case of (6.3) from the Euclidean case is again an analytic variety of codimension ≥ d and thus has measure zero w.r.t.µ × • • • × µ.Forming Zg t as above, it is a C N +1 perturbation of Z g 0 t and hence the conormal bundle N * Z g t is a C N perturbation of (6.5).As a result, for the same choices of partitions σ as in the Euclidean case, the canonical relations C σ t in the Riemannian case are C N perturbations of the C σ t analyzed in the previous section.Since C N perturbations of submersions are submersions, this means the same L 2 -Sobolev estimates hold, yielding nonempty interior of ∆diag g (E) for the same lower bounds on dim(E) (3.1) as in the Euclidean case.

Figure 1 .
Figure 1.A sketch of how one could view ∆ diag (E).
us to parametrize the open subset N * Zt ⊂ N * Z t as