Falconer Distance Research Articles

On restricted Falconer distance sets

Abstract 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 $$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$ for a compact $E\subset \mathbb {R}^d$ and $k\ge 3$ . We show that $\Delta ^{\mathrm{diag}}(E)$ has non-empty interior if the Hausdorff dimension of E satisfies (0.1) $$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$ 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.

Falconer type functions in three variables

Let f∈R[x,y,z] be a quadratic polynomial that depends on each variable and that does not have the form g(h(x)+k(y)+l(z)). Let A,B,C be compact sets in R. Suppose that dimH⁡(A)+dimH⁡(B)+dimH⁡(C)>2, then we prove that the image set f(A,B,C) is of positive Lebesgue measure. This dimensional condition is best possible in general. Our proof is based on a result due to Eswarathasan, Iosevich, and Taylor (Advances in Mathematics, 2011), and a combinatorial argument from the finite field model. As a consequence, we improve a result of Iosevich and Liu (2016) on the Falconer distance problem for Cartesian product sets in three dimensions.

Distribution of pinned distance trees in the plane [formula omitted

The recent result of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set A⊂R2, if the Hausdorff dimension of A is greater than 54, then the distance set Δ(A) has positive Lebesgue measure. With a completely different approach, Murphy, Petridis, Pham, Rudnev, and Stevens (2022) proved the prime field version of this result, namely, for E⊂Fp2 with |E|≫p5/4, there exist many points x∈E such that the number of distinct distances from x is at least cp. The main purpose of this paper is to prove extensions in the more general structure of pinned trees. The case of distances on small sets will also be addressed in this paper.

A Mattila-Sjölin theorem for triangles

We show for a compact set E⊂Rd, d≥4, that if the Hausdorff dimension of E is larger than 23d+1, then the set of congruence classes of triangles formed by triples of points of E has nonempty interior. Here we understand the set of congruence classes of triangles formed by triples of points of E as the setΔtri(E)={(t,r,α):|x−z|=t,|y−z|=r and Δtri(E)={,α=α(x,z,y),x,y,z∈E}, where α(x,z,y) denotes the angle formed by x, y and z, centered at z. This extends the Mattila-Sjölin theorem that establishes a non-empty interior for the distance set instead of the set of congruence classes of triangles. These theorems can be thought of as refinements and extensions of the statements in the well known Falconer distance problem.

Microlocal decoupling inequalities and the distance problem on Riemannian manifolds

We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth--Iosevich--Ou--Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran--Hickman--Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.

On the k-resultant modulus set problem on varieties over finite fields

Let [Formula: see text] be a regular variety, [Formula: see text] be an integer and [Formula: see text]. Covert et al. proved the following generalization of the Erdős–Falconer distance problem that if [Formula: see text], then we have [Formula: see text] In this paper, we provide improvements and extensions of their result.

Uniform distribution and geometric incidence theory

A celebrated unit distance conjecture due to Erdős says that the unit distances cannot arise more than Cϵn1+ϵ times (for any ϵ>0) among n points in the Euclidean plane (see e.g. [8] and the references contained therein). In three dimensions, the conjectured bound is Cn43 (see e.g. [6] and [11]). In dimensions four and higher, this problem, in its general formulation, loses meaning because the Lens example shows that one can construct a set of n points in dimension 4 and higher where the unit distance arises ≈n2 times (see e.g. [1]). However, the Lens example is one-dimension in nature, which raises the possibility that the unit distance conjecture is still quite interesting in higher dimensions under additional structural assumptions on the point set. This point of view was explored in [4], [5], [7] and has led to some interesting connections between the unit distance problem and its continuous counterparts, especially the Falconer distance conjecture ([3]). In this paper, we study the unit distance problem and its variants under the assumption that the underlying family of point sets is uniformly distributed. We prove several incidence bounds in this setting and clarify some key properties of uniformly distributed sequences in the context of incidence problems in combinatorial geometry.

On the pinned distances problem in positive characteristic

We study the Erdős–Falconer distance problem for a set A ⊂ F 2 $A\subset \mathbb {F}^2$ , where F $\mathbb {F}$ is a field of positive characteristic p $p$ . If F = F p $\mathbb {F}=\mathbb {F}_p$ and the cardinality | A | $|A|$ exceeds p 5 / 4 $p^{5/4}$ , we prove that A $A$ determines an asymptotically full proportion of the feasible p $p$ distances. For small sets A $A$ , namely when | A | ⩽ p 4 / 3 $|A|\leqslant p^{4/3}$ over any F $\mathbb {F}$ , we prove that either A $A$ determines ≫ | A | 2 / 3 $\gg |A|^{2/3}$ distances, or A $A$ lies on an isotropic line. For both large and small sets, the results proved are in fact for pinned distances.

Dimension of divergence set of the wave equation

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence limt→0eit−Δf(x)=f(x) fails when f∈Hs(Rd). We especially prove the conjecture raised by Barceló et al. (2011) for d=3, and improve the previous results in higher dimensions d≥4. We also show that a Strichartz type estimate for f→eit−Δf with the measure dtdμ(x) is essentially equivalent to the estimate for the spherical average of μ̂ which has been extensively studied for the Falconer distance set problem. The equivalence provides shortcuts to the recent results due to Liu (2019) and Rogers (2018).

Weighted restriction estimates and application to Falconer distance set problem

We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the Fourier transform of fractal measures, and therefore improve the results for the Falconer distance set conjecture in three and higher dimensions.

A sharp exponent on sum of distance sets over finite fields

We study a variant of the Erdős–Falconer distance problem in the setting of finite fields. More precisely, let E and F be sets in $$\mathbb {F}_q^d$$ , and $$\Delta (E), \Delta (F)$$ be corresponding distance sets. We prove that if $$|E||F|\ge Cq^{d+\frac{1}{3}}$$ for a sufficiently large constant C, then the set $$\Delta (E)+\Delta (F)$$ covers at least a half of all distances. Our result in odd dimensional spaces is sharp up to a constant factor. When E lies on a sphere in $${\mathbb {F}}_q^d,$$ it is shown that the exponent $$d+\frac{1}{3}$$ can be improved to $$d-\frac{1}{6}.$$ Finally, we prove a weak version of the Erdős–Falconer distance conjecture in four-dimensional vector spaces for multiplicative subgroups over prime fields. The novelty in our method is a connection with additive energy bounds of sets on spheres or paraboloids.

Erdös--Falconer Distance Problem under Hamming Metric in Vector Spaces over Finite Fields

For a subset $I\subseteq \mathbb{F}_{q}^{n}$, let $\Delta(I)$ be the set of distances determined by the elements of $I.$ The Erdös--Falconer distance problem in $\mathbb{F}_{q}^{n}$ asks for a threshold on the cardinality $|I|$ so that $\Delta(I)$ contains a positive proportion of the whole distance set. In this paper, we consider the analogous question under Hamming distance, which is the most important metric in coding theory. When $q\geqslant 4$ is a fixed prime power and $n$ goes to infinity, our main result shows that, for arbitrary positive proportion $\alpha,$ we can find $\alpha n$ distinct Hamming distances in $\Delta(I)$ if $|I|>q^{(1-\beta)\cdot n},$ where $\beta$ is a positive number depending on $\alpha.$ Unlike using Fourier analytical method as usual, our main tools include the celebrated dependent random choice and some results from additive number theory and coding theory. Hence our bound is much smaller than the previously known bound which was obtained by Fourier analytic machinery.

Dimensional lower bounds for Falconer type incidence theorems

Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2 \leq k \leq d$ and is obtained through counting simplices in a standard lattice using results of the Gauss circle problem. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We also establish a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing an example of Mattila. Finally, we prove a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for triangles in a convex setting in every dimension greater than $3$. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

On the Hausdorff dimension of pinned distance sets

We prove that if $A$ is a Borel set in the plane of equal Hausdorff and packing dimension $s>1$, then the set of pinned distances $\{ |x-y|:y\in A\}$ has full Hausdorff dimension for all $x$ outside of a set of Hausdorff dimension $1$ (in particular, for many $x\in A$). This verifies a strong variant of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint $s=1$.

Group actions, the Mattila integral and applications

The Mattila integral, $$ {\mathcal M}(\mu)=\int {\left( \int_{S^{d-1}} {|\widehat{\mu}(r \omega)|}^2 d\omega \right)}^2 r^{d-1} dr,$$ developed by Mattila, is the main tool in the study of the Falconer distance problem. In this paper, with a very simple argument, we develop a generalized version of the Mattila integral. Our first application is to consider the product of distances $$(\Delta(E))^k= \left\{\prod_{j=1}^k |x^j-y^j|: x^j, y^j\in E\right\} $$ and show that when $d\geq 2$, $(\Delta(E))^k$ has positive Lebesgue measure if $\dim_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{4k-1}$. Another application is, we prove for any $E,F,H\subset\mathbb{R}^2$, $\dim_{\mathcal{H}}(E)+\dim_{\mathcal{H}}(F)+\dim_{\mathcal{H}}(H)>4$, the set $$E\cdot(F+H)=\{x\cdot(y+z): x\in E, y\in F, z\in H\}$$ has positive Lebesgue.

An L2-identity and pinned distance problem

Let $\mu$ be a Frostman measure on $E\subset\mathbb{R}^d$. The spherical average decay $$\int_{S^{d-1}}|\widehat{\mu}(r\omega)|^2\,d\omega\lesssim r^{-\beta} $$ was originally used to attack Falconer distance conjecture, via Mattila's integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average decay implies the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres-Schlag's result on pinned distance problem significantly. The idea is to reduce the pinned distance problem to an integral where spherical averages apply. The key ingredient is the following identity. Using a group action argument, we show that for any Schwartz function $f$ on $\mathbb{R}^d$ and any $x\in\mathbb{R}^d$, $$\int_0^\infty |\omega_t*f(x)|^2\,t^{d-1}dt\,=\int_0^\infty|\widehat{\omega_r}*f(x)|^2\,r^{d-1}dr,$$ where $\omega_r$ is the normalized surface measure on $r S^{d-1}$. An interesting remark is that the right hand side can be easily seen equal to $$c_d\int\left|D_x^{-\frac{d-1}{2}}e^{-2\pi i t\sqrt{-\Delta}}f(x)\right|^2\,dt=c_d'\int\left|D_x^{-\frac{d-2}{2}}e^{2\pi i t\Delta}f(x)\right|^2\,dt.$$ An alternative derivation of Mattila's integral via group actions is also given in the Appendix.

The analog of the Erdös distance problem in finite fields

In this paper, we give a proof of the result of Iosevich and Rudnev [Erdös distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007) 6127–6142] on the analog of the Erdös–Falconer distance problem in the case of a finite field of characteristic [Formula: see text], where [Formula: see text] is an odd prime, without using estimates for Kloosterman sums. We also address the case of characteristic 2.

The k-resultant modulus set problem on algebraic varieties over finite fields

We study the k-resultant modulus set problem in the d-dimensional vector space Fqd over the finite field Fq with q elements. Given E⊂Fqd and an integer k≥2, the k-resultant modulus set, denoted by Δk(E), is defined asΔk(E)={‖x1±x2±⋯±xk‖∈Fq:xj∈E, j=1,2,…,k}, where ‖α‖=α12+⋯+αd2 for α=(α1,…,αd)∈Fqd. In this setting, the k-resultant modulus set problem is to determine the minimal cardinality of E⊂Fqd such that Δk(E)=Fq or Fq⁎. This problem is an extension of the Erdős–Falconer distance problem. In particular, we investigate the k-resultant modulus set problem with the restriction that the set E⊂Fqd is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof.

Falconer distance problem, additive energy and Cartesian products

A celebrated result due to Wolff says if $E$ is a compact subset of ${\Bbb R}^2$, then the Lebesgue measure of the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ is positive if the Hausdorff dimension of $E$ is greater than $\frac{4}{3}$. In this paper we improve the $\frac{4}{3}$ barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's $\frac{d}{2}+\frac{1}{3}$ exponent to $\frac{d^2}{2d-1}$. The proof uses a combination of Fourier analysis and additive comibinatorics.

A group-theoretic viewpoint on Erdős–Falconer problems and the Mattila integral

We obtain nontrivial exponents for Erdős–Falconer type point configuration problems. Let T_k(E) denote the set of distinct congruent k -dimensional simplices determined by (k+1) -tuples of points from E . For 1 \le k \le d , we prove that there exists a t_{k,d} < d such that, if E \subset {\mathbb R}^d , d \ge 2 , with \mathrm {dim}_{{\mathcal H}}(E)>t_{k,d} , then the {k+1 \choose 2} -imensional Lebesgue measure of T_k(E) is positive. Results of this type were previously obtained for triangles in the plane (k=d=2) in [8] and for higher k and d in [7]. We improve upon those exponents, using a group action perspective, which also sheds light on the classical approach to the Falconer distance problem.