arXiv subjects

Francisco Romero Acosta

Publications and source records attributed to Francisco Romero Acosta.

7 recordsLinked to original sources

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.

math.CA↗

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.

math.CA↗

arXiv2024

A Mattila-Sjölin theorem for simplices in low dimensions

In this paper we show that if a compact set $E \subset \mathbb{R}^d$, $d \geq 3$, has Hausdorff dimension greater than $\frac{(4k-1)}{4k}d+\frac{1}{4}$ when $3 \leq d<\frac{k(k+3)}{(k-1)}$ or $d- \frac{1}{k-1}$ when $\frac{k(k+3)}{(k-1)} \leq d$, then the set of congruence class of simplices with vertices in $E$ has nonempty interior. By set of congruence class of simplices with vertices in $E$ we mean $$Δ_{k}(E) = \left \{ \vec{t} = (t_{ij}) : |x_i-x_j|=t_{ij} ; \ x_i,x_j \in E ; \ 0\leq i < j \leq k \right \} \subset \mathbb{R}^{\frac{k(k+1)}{2}}$$ where $2 \leq k <d$. This result improves our previous work in the sense that we now can obtain a Hausdorff dimension threshold which allow us to guarantee that the set of congruence class of triangles formed by triples of points of $E$ has nonempty interior when $d=3$ as well as extending to all simplices. The present work can be thought of as an extension of the Mattila-Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.

math.CA↗

arXiv2023

VC-Dimension and Distance Chains in $\mathbb{F}_q^d$

Given a domain $X$ and a collection $\mathcal{H}$ of functions $h:X\to \{0,1\}$, the Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions $\mathcal{H}_t^{'2}(E): \mathbb{F}_q^2\to \{0,1\}$, corresponding to indicator functions of circles centered at points in a subset $E\subseteq \mathbb{F}_q^2$. They showed that when $|E|$ is large enough, the VC-dimension of $\mathcal{H}_t^{'2}(E)$ is the same as in the case that $E = \mathbb F_q^2$. We study a related hypothesis class, $\mathcal{H}_t^d(E)$, corresponding to intersections of spheres in $\mathbb{F}_q^d$, and ask how large $E\subseteq \mathbb{F}_q^d$ needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever $|E|\geq C_dq^{d-1/(d-1)}$ for $d\geq 3$, the VC-dimension of $\mathcal{H}_t^d(E)$ is as large as possible. We get a slightly stronger result if $d=3$: this result holds as long as $|E|\geq C_3 q^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq C_2 q^{7/4}$.

math.CO↗

arXiv2023

VC-Dimension of Hyperplanes over Finite Fields

Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where $h_y$ is the indicator function of the set $\{x\in E: x\cdot y=t\}$. Two of the authors, with Maxwell Sun, showed in the case $d=3$ that if $|E|\geq Cq^{\frac{11}{4}}$ and $q$ is sufficiently large, then the VC-dimension of $\mathcal{H}_t(E)$ is 3. In this paper, we generalize the result to arbitrary dimension and improve the exponent in the case $d=3$.

math.CO↗

arXiv2023

Distinct Angles and Angle Chains in Three Dimensions

In 1946, Erdős posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of $n$ points in the plane. The problem has since been explored along with many variants, including ones that extend it into higher dimensions. Less studied but no less intriguing is Erdős' distinct angle problem, which seeks to find point configurations in the plane that minimize the number of distinct angles. In their recent paper "Distinct Angles in General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum number of distinct angles in the plane in general position, which prohibits three points on any line or four on any circle. We consider the question of distinct angles in three dimensions and provide bounds on the minimum number of distinct angles in general position in this setting. We focus on pinned variants of the question, and we examine explicit constructions of point configurations in $\mathbb{R}^3$ which use self-similarity to minimize the number of distinct angles. Furthermore, we study a variant of the distinct angles question regarding distinct angle chains and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$ and $\mathbb{R}^3$.

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arXiv2022

A Mattila-Sjölin theorem for triangles

We show for a compact set $E \subset \mathbb{R}^d$, $d \geq 4$, that if the Hausdorff dimension of $E$ is larger than $\frac{2}{3}d+1$, 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 set $$Δ_{\text{tri}}(E) = \left \{ (t,r, α) : |x-z|=t, |y-z|=r \, \text{ and }\, α= α(x,z,y), \ x,y,z \in E \right \},$$ 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.

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