arXiv subjects
Publications and source records attributed to Doowon Koh.
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.
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.
For a set $E \subseteq \mathbb{F}_q^d$, the distance set is defined as $\Delta(E) := \{\|\mathbf{x} - \mathbf{y}\| : \mathbf{x}, \mathbf{y} \in E\}$, where $\|\cdot\|$ denotes the standard quadratic form. We investigate the Erd\H{o}s--Falconer distance problem within the flexible class of $(u, s)$--Salem sets introduced by Jonathan M. Fraser, with emphasis on the even case $u = 4$. By exploiting the exact identity between $\|\widehat{E}\|_4$ and the fourth additive energy $\Lambda_4(E)$, we prove that quantitative gains in $\Lambda_4(E)$ force the existence of many distances. In particular, for a $(4, s)$--Salem set $E\subset \mathbb{F}_q^d$ with $d \geq 2$, if \[ |E|\gg q^{\min\left\{\frac{d+2}{4s+1}, \frac{d+4}{8s}\right\}}, \] then $E$ determines a positive proportion of all distances. This strictly improves Fraser's threshold of $\frac{d}{4s}$ and the Iosevich-Rudnev bound of $q^{\frac{d+1}{2}}$ in certain parameter ranges. As applications, we obtain improved thresholds for multiplicative subgroups and sets on arbitrary varieties, and establish a sharp incidence bound for Salem sets that is of independent interest in incidence geometry. Moreover, our methods give sharp lower bounds for the number of distinct distances determined by two different sets. We also propose a unified conjecture for $(4, s)$--Salem sets that reconciles known bounds and pinpoints the odd-dimensional sphere regime: in odd dimensions $d \geq 3$, the often-cited $\frac{d-1}{2}$ threshold does not follow without additional structures. This provides a clear picture of the spherical distance conjecture.
Let $d\ge3$ and $\mathbb{F}_q^{\,d}$ be the $d$-dimensional vector space over a finite field of order $q$, where $q$ is an odd prime power. Let $X_\pi$ be the set of lines through the origin intersecting the slice $\pi\cap S^{d-1}$, where $\pi=\{x_d=\lambda\}$ and $S^{d-1}=\{x:\|x\|=1\}$. For $E\subset\mathbb{F}_q^{\,d}$ and $N\ge1$, we study the exceptional sets \[ T_1(X_\pi,E,N)=\bigl\{V\in X_\pi:\ |\pi_V(E)|\le N\bigr\},\qquad T_2(X_\pi,E,N)=\bigl\{V\in X_\pi:\ |\pi_{V^\perp}(E)|\le N\bigr\}, \] with their respective natural ranges of $N$. Using discrete Fourier analysis together with restriction/extension estimates for cone and sphere-type quadrics over finite fields, we obtain sharp upper bounds (up to constant factors) for $\lvert T_1\rvert$ and $\lvert T_2\rvert$, with separate analyses for the cases $\lambda \in \{0, \pm 1\}$. The bounds exhibit arithmetic-geometric dichotomies absent in the full Grassmannian: the quadratic character of $\lambda^{2}-1$ and the parity of $d$ determine the size of the exceptional sets. As an application, when $|E|\ge q$, there exists a positive proportion of elements $\mathbf{y}\in \pi\cap S^{d-1}$ such that the pinned dot-product sets $\{\mathbf{y}\cdot \mathbf{x}\colon \mathbf{x}\in E\}$ have cardinality $\Omega(q)$. We further study analogous families arising from the spheres of radii $0$ and $-1$, and, by combining the results, recover the known estimates for projections over the full Grassmannian, complementing a result of Chen (2018).
We bound the number of incidences between points and spheres in finite vector spaces by bounding the sum of the number of points in the pairwise intersections of the spheres. We obtain new incidence bounds that are interesting when the number of spheres is not too large. Our approach also leads to an elementary proof of the Iosevich-Rudnev bound on the Erd\H{o}s-Falconer distance problem in odd dimensions.
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 $\mathbb{F}_q$.
We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other ideas.
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.
Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\mathbb F_q$ and $E\subset \mathbb F_q^d$, we define $W(r)$ as the number of quadruples $(x,y,z,w)\in E^4$ such that $ Q(x-y)/Q(z-w)=r,$ where $Q$ is a non-degenerate quadratic form in $d$ variables over $\mathbb F_q.$ When $Q(α)=\sum_{i=1}^d α_i^2$ with $α=(α_1, \ldots, α_d)\in \mathbb F_q^d,$ Pham (2022) recently used the machinery of group actions and proved that if $E\subset \mathbb F_q^2$ with $q\equiv 3 \pmod{4}$ and $|E|\ge C q$, then we have $W(r)\ge c |E|^4/q$ for any non-zero square number $r \in \mathbb F_q,$ where $C$ is a sufficiently large constant, $ c$ is some number between $0$ and $1,$ and $|E|$ denotes the cardinality of the set $E.$ In this article, we improve and extend Pham's result in two dimensions to arbitrary dimensions with general non-degenerate quadratic distances. As a corollary of our results, we also generalize the sharp results on the Falconer type problem for the quotient set of distance set due to the first two authors and Parshall (2019). Furthermore, we provide improved constants for the size conditions of the underlying sets. The key new ingredient is to relate the estimate of the $W(r)$ to a quadratic homogeneous variety in $2d$-dimensional vector space. This approach is fruitful because it allows us to take advantage of Gauss sums which are more handleable than the Kloosterman sums appearing in the standard distance type problems.
In this paper, we prove an extension theorem for spheres of square radii in $\mathbb{F}_q^d$, which improves a result obtained by Iosevich and Koh (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restriction theorem due to the authors and Lee (2022). Applications on the distance problems will also be discussed.
The purpose of this paper is to introduce and study the following graph theoretic paradigm. Let $$T_Kf(x)=\int K(x,y) f(y) dμ(y),$$ where $f: X \to {\Bbb R}$, $X$ a set, finite or infinite, and $K$ and $μ$ denote a suitable kernel and a measure, respectively. Given a connected ordered graph $G$ on $n$ vertices, consider the multi-linear form $$ Λ_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) dμ(x^l), $$ where ${\mathcal E}(G)$ is the edge set of $G$. Define $Λ_G(p_1, \ldots, p_n)$ as the smallest constant $C>0$ such that the inequality $$ Λ_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, μ)}$$ holds for all non-negative real-valued functions $f_i$, $1\le i\le n$, on $X$. The basic question is, how does the structure of $G$ and the mapping properties of the operator $T_K$ influence the sharp exponents. In this paper, this question is investigated mainly in the case $X={\Bbb F}_q^d$, the $d$-dimensional vector space over the field with $q$ elements, and $K(x^i,x^j)$ is the indicator function of the sphere evaluated at $x^i-x^j$. This provides a connection with the study of $L^p$-improving measures and distance set problems.
In this paper we study the restriction estimate for the flat disk over finite fields. Mockenhaupt and Tao initially studied this problem but their results were addressed only for dimensions $n=4,6$. We improve and extend their results to all dimensions $n\geq 6$. More precisely, we obtain the sharp $L^2 \to L^r$ estimates, which cannot be proven by applying the usual Stein-Tomas argument over a finite field even with the optimal Fourier decay estimate on the flat disk. One of main ingredients is to discover and analyze an explicit form of the Fourier transform of the surface measure on the flat disk. In addition, based on the recent results on the restriction estimates for the paraboloids, we address improved restriction estimates for the flat disk beyond the $L^2$ restriction estimates.
Let F_q be a finite field with odd q elements. In this article, we prove that if E \subseteq \mathbb F_q^d, d\ge 2, and |E|\ge q, then there exists a set Y \subseteq \mathbb F_q^d with |Y|\sim q^d$ such that for all y\in Y, the number of distances between the point y and the set E is similar to the size of the finite field \mathbb F_q. As a corollary, we obtain that for each set E\subseteq \mathbb F_q^d with |E|\ge q, there exists a set Y\subseteq \mathbb F_q^d with |Y|\sim q^d so that any set E\cup \{y\} with y\in Y determines a positive proportion of all possible distances. An averaging argument and the pigeonhole principle play a crucial role in proving our results.
Let $\mathbb{F}_q$ be a finite field of order $q$. Iosevich and Rudnev (2005) proved that for any set $A\subset \mathbb{F}_q^d$, if $|A|\gg q^{\frac{d+1}{2}}$, then the distance set $Δ(A)$ contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be $\frac{d}{2}$ in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture is still wide open in higher dimensions. To fill the gap, we need to understand more about the structures of the distance sets, the main purpose of this paper is to provide some structural theorems on the distribution of square and non-square distances.
Let $ϕ(x, y)\colon \mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}$ be a function. We say $ϕ$ is a Mattila--Sjölin type function of index $γ$ if $γ$ is the smallest number satisfying the property that for any compact set $E\subset \mathbb{R}^d$, $ϕ(E, E)$ has a non-empty interior whenever $\dim_H(E)>γ$. The usual distance function, $ϕ(x, y)=|x-y|$, is conjectured to be a Mattila--Sjölin type function of index $\frac{d}{2}$. In the setting of finite fields $\mathbb{F}_q$, this definition is equivalent to the statement that $ϕ(E, E)=\mathbb{F}_q$ whenever $|E|\gg q^γ$. The main purpose of this paper is to prove the existence of such functions with index $\frac{d}{2}$ in the vector space $\mathbb{F}_q^d$.
Let $\mathbb{F}_q$ be a finite field of order $q$. In this paper, we study the distribution of rectangles in a given set in $\mathbb{F}_q^2$. More precisely, for any $0<δ\le 1$, we prove that there exists an integer $q_0=q_0(δ)$ with the following property: if $q\ge q_0$ and $A$ is a multiplicative subgroup of $\mathbb{F}^*_q$ with $|A|\ge q^{2/3}$, then any set $S\subset \mathbb{F}_q^2$ with $|S|\ge δq^2$ contains at least $\gg \frac{|S|^4|A|^2}{q^5}$ rectangles with side-lengths in $A$. We also consider the case of rectangles with one fixed side-length and the other in a multiplicative subgroup $A$.
In this paper, we prove a new point-sphere incidence bound in vector spaces over finite fields. More precisely, let $P$ be a set of points and $S$ be a set of spheres in $\mathbb{F}_q^d$. Suppose that $|P|, |S|\le N$, we prove that the number of incidences between $P$ and $S$ satisfies \[I(P, S)\le N^2q^{-1}+q^{\frac{d-1}{2}}N,\] under some conditions on $d, q$, and radii. This improves the known upper bound $N^2q^{-1}+q^{\frac{d}{2}}N$ in the literature. As an application, we show that for $A\subset \mathbb{F}_q$ with $q^{1/2}\ll |A|\ll q^{\frac{d^2+1}{2d^2}}$, one has \[\max \left\lbrace |A+A|,~ |dA^2|\right\rbrace \gg \frac{|A|^d}{q^{\frac{d-1}{2}}}.\] This improves earlier results on this sum-product type problem over arbitrary finite fields.
The first purpose of this paper is to solve completely the finite field cone restriction conjecture in four dimensions with $-1$ non-square. The second is to introduce a new approach to study incidence problems via restriction theory. More precisely, using the cone restriction estimates, we will prove sharp point-sphere incidence bounds associated with complex-valued functions for sphere sets of small size. Our incidence bounds with a specific function improve significantly a result given by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev.