Parallelograms and the VC-dimension of the distance sets

In this paper, we study the distribution of parallelograms and rhombi in a given set in the plane over arbitrary finite fields $\mathbb{F}_q^2$. As an application, we improve a recent result due to Fitzpatrick, Iosevich, McDonald, and Wyman (2021) on the Vapnik-Chervonenkis dimension of the induced distance graph. Our proofs are based on the discrete Fourier analysis.Comment: 9 page

Distinct distances on regular varieties over finite fields

In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset $\mathcal{E}$ in a regular variety satisfies $|\mathcal{E}|\gg q^{\frac{d-1}{2}+\frac{1}{k-1}}$, then $\Delta_{k, F}(\mathcal{E})\supseteq \mathbb{F}_q\setminus \{0\}$ for some certain families of polynomials $F(\mathbf{x})\in \mathbb{F}_q[x_1, \ldots, x_d]$

Pinned-base simplex, a Furstenberg type problem, and incidences in finite vector spaces

In this paper we prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in $\mathbb{F}_q^2$. More precisely, for $A, B, C\subset \mathbb{F}_q^2$, if $|A||B||C|^{1/2}\gg q^4$, then for any $\lambda\in \mathbb{F}_q\setminus \{0\}$, the number of congruence classes of triangles with vertices in $A\times B\times C$ and one side-length $\lambda$ is at least $\gg q^2$. As a consequence, the number of congruence classes of triangles with vertices in $A\times B\times C$ is at least $\gg q^3$. The main ingredients in our proof are a recent incidence bound between points and rigid motions due to the author and Semin Yoo (2023) and a result on a Furstenberg type problem. When three sets are the same, we give a unified and new proof for all the best current results due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2017) and McDonald (2020). The novelty of this approach is to present an application of results on the number of $k$-rich rigid motions in studying the distribution of simplex. A number of related questions will be also addressed in this paper.Comment: 21 page

Group action and L2L^2-norm estimates of geometric problems

In 2017, by using the group theoretic approach, Bennett, Hart, Iosevich, Pakianathan, and Rudnev obtained a number of results on the distribution of simplices and sum-product type problems. The main purpose of this paper is to give a series of new applications of their powerful framework, namely, we focus on the product and quotient of distance sets, the $L^2$-norm of the direction set, and the $L^2$-norm of scales in difference sets.Comment: v2: accepted versio

Distinct spreads in vector spaces over finite fields

In this short note, we study the distribution of spreads in a point set $\mathcal{P} \subseteq \mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $\varepsilon > 0$, if $|\mathcal{P}| \geq (1+\varepsilon) q^{\lceil d/2 \rceil}$, then $\mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $\mathcal{P} \subset \mathbb{F}_q^d$ of size $|\mathcal{P}| = q^{\lceil d/2 \rceil}$ that determine at most one spread