arXiv subjects
Publications and source records attributed to Hunseok Kang.
In this paper, we study the $\ell^p\to \ell^r$ estimates for the $S$-operator arising in restriction problems for spheres over finite fields. We establish a necessary and sufficient condition for the boundedness of the $S$-operator. Furthermore, we investigate this problem under certain restrictions on test functions. In particular, we address the sharp results when test functions are restricted to radial functions.
Let $\mathbb{F}_q^d$ be a $d$-dimensional vector space over a finite field $\mathbb{F}_q$ with $q$ elements. For $x\in \mathbb{F}_q^d$, let $\|x\| = x_1^2+\dots+x_d^2$. By abuse of terminology, we shall call $\|\cdot\|$ a norm on $\mathbb{F}_q^d$. For a subset $E\subset \mathbb{F}_q^d$, let $\Delta(E)$ be the distance set on $E$ defined as $\Delta(E):=\{\|x-y\| : x, y \in E \}$. The Mattila-Sj\"olin problem seeks the smallest exponent $\alpha>0$ such that $\Delta(E) =\mathbb{F}_q$ for all subsets $E \subset \mathbb{F}_q^d$ with $|E| \geq Cq^\alpha$. In this article, we consider this problem for a variant of this norm, which generates a smaller distance set than the norm $\|\cdot\|.$ Namely, we replace the norm $\|\cdot\|$ by the so-called $k$-norm $(1 \leq k \leq d)$, which can be viewed as a kind of deformation of $\|\cdot\|$. To derive our result on the Mattila-Sj\"olin problem for the $k$-norm, we use a combinatorial method to analyze various summations arising from the discrete Fourier machinery. Even though our distance set is smaller than the one in the Mattila-Sj\"olin problem, for some $k$ we still obtain the same result as that of Iosevich and Rudnev (2007), which deals with the Mattila-Sj\"olin problem. Furthermore, our result is sharp in all odd dimensions.
We extend Raimi's classical partition theorem to the continuous setting of the circle and $n$-dimensional torus. Building on recent work of Hegyv\'ari, Pach, and Pham in finite groups, we prove that there exist measurable partitions of the $n$-dimensional torus $\mathbb{T}^n$ with the property that for any finite measurable cover, some translated part of the cover has positive measure intersection with every partition element. Our proof adapts combinatorial arguments from the finite setting using measure-theoretic techniques and slicing arguments in product spaces.
In this paper, we study the cardinality of the distance set $\Delta(A, B)$ determined by two subsets $A$ and $B$ of the $d$-dimensional vector space over a finite field $\mathbb{F}_q$. Assuming that $A$ or $B$ lies in a $k$-coordinate plane up to translations and rotations, we prove that if $|A||B| > 2q^d$, then $|\Delta(A, B)| > q/2$, where $|\Delta(A, B)|$ denotes the number of distinct distances between elements of $A$ and $B$. In particular, we show that our result recovers the sharp $(d+1)/2$ threshold for the Erd\H{o}s-Falconer distance problem in odd dimensions, where distances are determined by a single set. As an application, we also obtain an improved result on the Box distance problem posed by Borges, Iosevich, and Ou, in the case where $2$ is a square in $\mathbb{F}_q$.
In this paper, we introduce a new operator, $\mathcal{S}$, which is closely related to the restriction problem for spheres in $\mathbb{F}_q^d$, the $d$-dimensional vector space over the finite field $\mathbb{F}_q$ with $q$ elements. The $\mathcal{S}$ operator is considered as a specific operator that maps functions on $\mathbb{F}_q^d$ to functions on $\mathbb{F}_q^{d+1}$. We explore a relationship between the boundedness of the $\mathcal{S}$ operator and the restriction estimate for spheres in $\mathbb{F}_q^d$. Consequently, using this relationship, we prove that the $L^2$ restriction conjectures for spheres hold in all dimensions when the test functions are restricted to homogeneous functions of degree zero.
We study L^p-L^r restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension $d$ is even, then it is conjectured that the L^{(2d+2)/(d+3)}-L^2 Stein-Tomas restriction result can be improved to the L^{(2d+4)/(d+4)}-L^2 estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured L^p-L^2 restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in $d$ dimensions and those for homogeneous varieties in (d+1)dimensions.