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Yumeng Ou

Publications and source records attributed to Yumeng Ou.

At least 19 recordsLinked to original sources
arXiv2026

From weighted paraboloid restriction to $k$-stars and distance graphs

In this paper, we study pinned $k$-star distance sets associated to compact subsets of $\mathbb{R}^n$, $n\geq 2$. For pins $x_1,\dots,x_k\in E$, the pinned $k$-star distance set is \[ \Delta_{x_1,\dots,x_k}^{k\text{-star}}(E) = \{(|x_1-x|,\dots,|x_k-x|):x\in E\}\subset\mathbb{R}^k. \] We obtain improved Hausdorff-dimension thresholds on $E$ guaranteeing that pinned $k$-star distance sets have positive $k$-dimensional Lebesgue measure. The main analytic input is a reformulation of the connection, first observed in \cite{IPPS22}, between $k$-stars in $\mathbb{R}^n$ and pinned dot products on the paraboloid in $\mathbb{R}^{n+1}$. In our framework, $L^2(\mathbb{R}^k)$ estimates for the densities of pinned $k$-star distance measures are reduced to a weighted Fourier extension estimate for the paraboloid whose weight is defined explicitly in terms of Frostman measures on $E$. For $1\leq k \alpha_{+}(n,k):=\frac{n^2+nk+k}{2n+1}=\frac{n+k-1}{2}+\frac14 +\frac{2k+1}{4(2n+1)}.\] Using the graph-building machinery of \cite{BFOPR2026}, our positive-measure results for $k$-stars can be used as building blocks for finite distance graph configurations with prescribed pins. As a consequence, we improve the best-known positive-measure thresholds for pinned $k$-simplices in every dimension $n\geq 3$ and for necklace graphs (cycles) in every dimension $n\geq 3$. We further prove nonempty interior results for $k$-stars. In the special case $k=1$, corresponding to the pinned nonempty interior of the distance set $\Delta_{x}(E)=\{|x-y|\colon y\in E\}$, we use a sharper argument to improve the pinned nonempty-interior thresholds of \cite{BFOP2026} in all dimensions $n\geq 4$.

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arXiv2026

On volume vectors determined by hypergraphs in thin subsets of Euclidean space

Generalizing the Falconer distance problem, the authors of this paper recently established the first non-trivial dimensional threshold for any distance graph in high enough of a dimension. The methods developed were flexible enough to generalize from the Euclidean distance to any two point configuration, conditional on results on $k$-stars for the two point configuration. A natural question emerges on what happens to configurations that take in more than two points. In this paper we consider a classic three point variant of the Falconer distance problem, namely that on areas of triangles and its generalizations to volumes of simplices. In this model case we develop two methods. One we call the Jacobian method which allows us, through Heron's formula, to leverage earlier results on distance graphs and obtains non-trivial thresholds for volume vectors determined by a wide range of hypergraphs of simplices. Even in the classic case of the volume of a single simplex this method yields the best known dimensional thresholds if the dimension is considerably bigger than the size of the simplex. We develop a conjecture that has connections to rigidity theory. The Jacobian method works best in high dimensions so in the case of areas of triangles in the plane, we refine the work of Shmerkin and Yavicoli, who recently resolved a conjecture for areas of triangles in the plane, and obtain building blocks from which we can get abundance of area vectors determined by certain hypergraphs of triangles, such as chains of triangles connected on edges or vertices. The results improve and extend existing results of Galo and McDonald as well as of Greenleaf, Iosevich and Taylor.

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arXiv2026

Falconer-type results for any finite graph with multiple pins

A generalization of the celebrated Falconer distance problem asks for a graph $G=(\mathcal{V},\mathcal{E})$, with vertex set $\mathcal{V}$ and edge set $\mathcal{E}$, how large the Hausdorff dimension of a compact set $E\subset \mathbb{R}^d$, $d\geq 2$, needs to be to guarantee that the distance graph $$ Δ^{G}(E):= \lbrace (|x_{i}-x_{j}|)_{(v_i,v_j)\in\mathcal{E}} : x_1,\ldots,x_{|\mathcal{V}|}\in E \rbrace $$ has positive $|\mathcal{E}|$-dimensional Lebesgue measure. Here we represent the edges in $\mathcal{E}$ as ordered pairs of vertices $(v_i,v_j)$ with $i k$, for any non-trivial graph $G$. Key ingredients for our result are identifying pinned stars as the right building blocks for a general graph as well as refining a Fubini type argument due to Taylor and the third named author. We further generalize this to graphs with multiple pins by introducing the $k$-admissibility of a graph, a generalization of $k$-degeneracy that takes pins into account, as well as by extending the Fubini argument to the multiple pinned setting. Not only do we obtain non-trivial results in high enough dimensions for any distance graph, but for particular graphs (such as cycles) our results are also strong and improve the previously best known results. Our methods extend to general two point configurations, contingent on results being available for the appropriate star building blocks.

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arXiv2025

Nonempty interior of pinned distance and tree sets

For a compact set $E\subset\mathbb{R}^d$, $d\geq 2$, consider the pinned distance set $\Delta^{y}(E)=\lbrace |x-y| : x\in E\rbrace$. Peres and Schlag showed that if the Hausdorff dimension of $E$ is bigger than $\frac{d+2}{2}$ with $d\geq 3$, then there exists a point $y\in E$ such that $\Delta^{y}(E)$ has nonempty interior. In this paper we obtain the first non-trivial threshold for this problem in the plane, improving on the Peres--Schlag threshold when $d=3$, and we extend the results to trees using a novel induction argument.

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arXiv2025

Incidence bounds related to circular Furstenberg sets

We prove bounds on approximate incidences between families of circles and families of points in the plane. As a consequence, we prove a lower bound for the dimension of circular $(u,v)$-Furstenberg sets, which is new for large $u$ and $v$.

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arXiv2024

Weighted Estimates of Singular Integrals and Commutators in the Zygmund Dilation Setting

The main purpose of this paper is to establish weighted estimates for singular integrals associated with Zygmund dilations via a discrete Littlewood--Paley theory, and then apply it to obtain the upper bound of the norm of commutators of such singular integrals with a function in the little bmo space associated with Zygmund dilations. Examples of such singular integrals associated with Zygmund dilations include a class of singular integrals studied by Ricci--Stein and Fefferman--Pipher, as well as a singular integral along a particular surface studied by Nagel--Wainger. We show that the lower bound of the norm of this commutator is not true for any singular integral in the class considered in Ricci--Stein and Fefferman--Pipher, but does in fact hold for the specific singular integral studied in Nagel--Wainger. In particular this implies that the family of singular integrals studied in these papers is not sufficiently general to contain the operator of Nagel--Wainger, which we show is of significance in this theory.

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arXiv2024

New improvement to Falconer distance set problem in higher dimensions

We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $Δ_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\dim_H (E) \in (\frac{d}{2} - \frac{1}{4} - \frac{3}{8d+4}, \frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimensions $d \ge 3$.

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arXiv2023

Sobolev smoothing estimates for bilinear maximal operators with fractal dilation sets

Given a hypersurface $S\subset \mathbb{R}^{2d}$, we study the bilinear averaging operator that averages a pair of functions over $S$, as well as more general bilinear multipliers of limited decay and various maximal analogs. Of particular interest are bilinear maximal operators associated to a fractal dilation set $E\subset [1,2]$; in this case, the boundedness region of the maximal operator is associated to the geometry of the hypersurface and various notions of the dimension of the dilation set. In particular, we determine Sobolev smoothing estimates at the exponent $L^2 \times L^2 \rightarrow L^2$ using Fourier-analytic methods, which allow us to deduce additional $L^p$ improving bounds for the operators and sparse bounds and their weighted corollaries for the associated multi-scale maximal functions. We also extend the method to study analogues of these questions for the triangle averaging operator and biparameter averaging operators. In addition, some necessary conditions for boundedness of these operators are obtained.

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arXiv2023

Weighted refined decoupling estimates and application to Falconer distance set problem

We prove some weighted refined decoupling estimates. As an application, we give an alternative proof of the following result on Falconer's distance set problem by the authors in a companion work: if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 4$, then there is a point $x\in E$ such that the pinned distance set $Δ_x(E)$ has positive Lebesgue measure. Aside from this application, the weighted refined decoupling estimates may be of independent interest.

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arXiv2023

A singular variant of the Falconer distance problem

In this paper we study the following variant of the Falconer distance problem. Let $E$ be a compact subset of ${\mathbb{R}}^d$, $d \ge 1$, and define $$ \Box(E)=\left\{\sqrt{{|x-y|}^2+{|x-z|}^2}: x,y,z \in E,\, y\neq z \right\}.$$ We shall prove using a variety of methods that if the Hausdorff dimension of $E$ is greater than $\frac{d}{2}+\frac{1}{4}$, then the Lebesgue measure of $\Box(E)$ is positive. This problem can be viewed as a singular variant of the classical Falconer distance problem because considering the diagonal $(x,x)$ in the definition of $\Box(E)$ poses interesting complications stemming from the fact that the set $\{(x,x): x \in E\}\subseteq \mathbb{R}^{2d}$ is much smaller than the sets for which the Falconer type results are typically established. We also prove a finite field variant of the Euclidean results for $\Box(E)$ and indicate both the similarities and the differences between the two settings.

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arXiv2022

Sparse bounds for the bilinear spherical maximal function

We derive sparse bounds for the bilinear spherical maximal function in any dimension $d\geq 1$. When $d\geq 2$, this immediately recovers the sharp $L^p\times L^q\to L^r$ bound of the operator and implies quantitative weighted norm inequalities with respect to bilinear Muckenhoupt weights, which seems to be the first of their kind for the operator. The key innovation is a group of newly developed continuity $L^p$ improving estimates for the single scale bilinear spherical averaging operator.

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arXiv2022

On the multiparameter Falconer distance problem

We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell\geq 1$ and $\mathbb{R}^{d}=\mathbb{R}^{d_1}\times\cdots \times\mathbb{R}^{d_\ell}$, $d_i\geq 2$. For any compact set $E\subset \mathbb{R}^{d}$ with Hausdorff dimension larger than $d-\frac{\min(d_i)}{2}+\frac{1}{4}$ if $\min(d_i) $ is even, $d-\frac{\min(d_i)}{2}+\frac{1}{4}+\frac{1}{4\min(d_i)}$ if $\min(d_i) $ is odd, we prove that the multiparameter distance set of $E$ has positive $\ell$-dimensional Lebesgue measure. A key ingredient in the proof is a new multiparameter radial projection theorem for fractal measures.

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arXiv2021

An improved result for Falconer's distance set problem in even dimensions

We show that if compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}$, where $d\geq 4$ is an even integer, then the distance set of $E$ has positive Lebesgue measure. This improves the previously best known result towards Falconer's distance set conjecture in even dimensions.

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arXiv2021

A cone restriction estimate using polynomial partitioning

We obtain improved Fourier restriction estimate for the truncated cone using the method of polynomial partitioning in dimension $n\geq 3$, which in particular solves the cone restriction conjecture for $n=5$, and recovers the sharp range for $3\leq n\leq 4$. The main ingredient of the proof is a $k$-broad estimate for the cone extension operator, which is a weak version of the $k$-linear cone restriction estimate for $2\leq k\leq n$.

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arXiv2020

Finite Point Configurations and the Regular Value Theorem in a Fractal setting

In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $E\subset \mathbb{R}^d$. The first problem concerns how the Lebesgue measure or the Hausdorff dimension of the finite point configuration set depends on that of $E$. In particular, we show that if a planar set has dimension exceeding $\frac{5}{4}$, then there exists a point $x\in E$ so that for each integer $k\geq2$, the set of "$k$-chains" with initial point at $x$ has positive Lebesgue measure. The second problem is a continuous analogue of the Erdős unit distance problem, which aims to determine the maximum number of times a point configuration with prescribed gaps can appear in $E$. For instance, given a triangle with prescribed sides and given a sufficiently regular planar set $E$ with Hausdorff dimension no less than $\frac{7}{4}$, we show that the dimension of the set of vertices in $E$ forming said triangle does not exceed $3\,\dim_{\mathcal{H}} (E)-3$. In addition to the Euclidean norm, we consider more general distances given by functions satisfying the so-called Phong-Stein rotational curvature condition. We also explore a number of examples to demonstrate the extent to which our results are sharp.

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arXiv2019

Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type

We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<\infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmund's $L{(\log L)^{\frac12}}$ inequality is the core of our lacunary multi-frequency projection proof. We use the Walsh setting to avoid extra complications in the arguments.

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