arXiv subjects
Publications and source records attributed to Doowon Koh.
Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough.
We prove that the finite field Fourier extension operator for the paraboloid is bounded from $L^2\to L^r$ for $r\geq \frac{2d+4}{d}$ in even dimensions $d\ge 8$, which is the optimal $L^2$ estimate. For $d=6$ we obtain the optimal range $r> \frac{2d+4}{d}=8/3$, apart from the endpoint. For $d=4$ we improve the prior range of $r>16/5=3.2$ to $r\geq 28/9=3.111\ldots$, compared to the conjectured range of $r\geq3$. The key new ingredient is improved additive energy estimates for subsets of the paraboloid.
Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. Given $k$ sets $E_j\subset \mathbb F_q^d$ for $j=1,2,\ldots, k$, the generalized $k$-resultant modulus set, denoted by $Δ_k(E_1,E_2, \ldots, E_k)$, is defined by $$ Δ_k(E_1,E_2, \ldots, E_k)=\left\{\|{\bf x}^1+{\bf x}^2+\cdots+{\bf x}^k\|\in \mathbb F_q:{\bf x}^j\in E_j,\, j=1,2,\ldots, k\right\},$$ where $\|{\bf y}\|={\bf y}_1^2+ \cdots + {\bf y}_d^2$ for ${\bf y}=({\bf y}_1, \ldots, {\bf y}_d)\in \mathbb F_q^d.$ We prove that if $\prod\limits_{j=1}^3 |E_j| \ge C q^{3\left(\frac{d+1}{2} -\frac{1}{6d+2}\right)}$ for $d=4,6$ with a sufficiently large constant $C>0$, then $|Δ_3(E_1,E_2,E_3)|\ge cq$ for some constant $0 0$ from the exponent.
We study the extension estimates for paraboloids in d-dimensional vector spaces over finite fields F_q with q elements. We use the connection between L^2 based restriction estimates and L^p\to L^r extension estimates for paraboloids. As a consequence, we improve the L^2\to L^r extension results obtained by A. Lewko and M. Lewko in even dimensions d\ge 6 and odd dimensions d=4\ell+3 for \ell \in \mathbb N. Our results extend the consequences for 3-D paraboloids due to M. Lewko to higher dimensions. We also clarifies conjectures on finite field extension problems for paraboloids.
We investigate the sharp L^p\to L^r estimates for the restricted averaging operator A_C over the cone C of the d-dimensional vector space F_q^d over the finite field F_q with q elements. The restricted averaging operator A_C for the cone C is defined by the relation that A_Cf=f\ast σ|_C, where σdenotes the normalized surface measure on the cone C, and f is a complex valued function on the space F_q^d with the normalized counting measure dx. In the previous work, the sharp boundedness of A_C was obtained in odd dimensions d\ge 3 but partial results were only given in even dimensions d\ge 4. In this paper we prove the optimal estimates in even dimensions d\ge 6 in the case when the cone C\subset F_q^d contains a d/2 dimensional subspace.
We study mapping properties of finite field k-plane transforms. Using geometric combinatorics, we do an elaborate analysis to recover the critical endpoint estimate. As a consequence, we obtain optimal L^p-L^r estimates for all k-plane transforms in the finite field setting. In addition, applying Holder's inequality to our results, we obtain an estimate for multilinear k-plane transforms.
We study $L^p\to L^r$ estimates for restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb F_q$ with $q$ elements. We observe properties of both the Fourier restriction operator and the averaging operator over $V\subset \mathbb F_q^d.$ As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions $d\ge2,$ and cones in odd dimensions $d\ge 3.$ In addition, when the variety $V$ is a cone lying in an even dimensional vector space over $\mathbb F_q$ and $-1$ is a square number in $\mathbb F_q$, we also obtain sharp estimates except for two endpoints.
We study the $k$-resultant modulus set problem in the $d$-dimensional vector space $\mathbb F_q^d$ over the finite field $\mathbb F_q$ with $q$ elements. Given $E\subset \mathbb F_q^d$ and an integer $k\ge 2$, the $k$-resultant modulus set, denoted by $Δ_k(E)$, is defined as $$ Δ_k(E)=\{\|x^1\pm x^2 \pm \cdots \pm x^k\|\in \mathbb F_q: x^j\in E, ~j=1,2,\ldots, k\},$$ where $\|α\|=α_1^2+\cdots+ α_d^2$ for $α=(α_1, \ldots, α_d) \in \mathbb F_q^d.$ In this setting, the $k$-resultant modulus set problem is to determine the minimal cardinality of $E\subset \mathbb F_q^d$ such that $Δ_k(E) = \mathbb F_q$ or $\mathbb{F}_q^*$. This problem is an extension of the Erdős-Falconer distance problem. In particular, we investigate the $k$-resultant modulus set problem with the restriction that the set $E\subset \mathbb F_q^d$ is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof.
For a set $E\subset \mathbb F_q^d$, we define the $k$-resultant magnitude set as $ Δ_k(E) =\{\|\textbf{x}_1 + \dots + \textbf{x}_k\|\in \mathbb F_q: \textbf{x}_1, \dots, \textbf{x}_k \in E\},$ where $\|\textbf{v}\|=v_1^2+\cdots+ v_d^2$ for $\textbf{v}=(v_1, \ldots, v_d) \in \mathbb F_q^d.$ In this paper we find a connection between a lower bound of the cardinality of the $k$-resultant magnitude set and the restriction theorem for spheres in finite fields. As a consequence, it is shown that if $E\subset \mathbb F_q^d$ with $|E|\geq C q^{\frac{d+1}{2}-\frac{1}{6d+2}},$ then $|Δ_3(E)|\geq c q$ for $d = 4$ or $d = 6$, and $|Δ_4(E)| \geq cq$ for even dimensions $d \geq 8.$ In addition, we prove that if $d\geq 8$ is even, and $|E|\geq C_\varepsilon ~q^{\frac{d+1}{2} - \frac{1}{9d -18} + \varepsilon}$ for $\varepsilon >0$, then $|Δ_3(E)|\geq c q.$
In this paper we study the cardinality of the dot product set generated by two subsets of vector spaces over finite fields. We notice that the results on the dot product problems for one set can be simply extended to two sets. Let E and F be subsets of the d-dimensional vector space F_q^d over a finite field F_q with q elements. As a new result, we prove that if E and F are subsets of the paraboloid and |E||F|\geq C q^d for some large C>1, then |Π(E,F)|\geq c q for some 0<c<1. In particular, we find a connection between the size of the dot product set and the number of lines through both the origin and a nonzero point in the given set E. As an application of this observation, we obtain more sharpened results on the generalized dot product set problems. The discrete Fourier analysis and geometrical observation play a crucial role in proving our results.
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.
We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we improve upon the results by Rainer Dietmann. In the case that one of the subsets is a product set, we obtain further improvement on the estimate.
In this paper we study the mapping properties of the averaging operator over a variety given by a system of homogeneous equations over a finite field. We obtain optimal results on the averaging problems over two dimensional varieties whose elements are common solutions of diagonal homogeneous equations. The proof is based on a careful study of algebraic and geometric properties of such varieties. In particular, we show that they are not contained in any hyperplane and are complete intersections. We also address partial results on averaging problems over arbitrary dimensional homogeneous varieties which are smooth away from the origin.
We study mapping properties of the averaging operator related to the variety $ V={x\in \mathbb F_q^d: Q(x)=0},$ where $Q(x)$ is a nondegenerate quadratic polynomial over a finite field $\mathbb F_q$ with $q$ elements. This paper is devoted to eliminating the logarithmic bound appearing in the paper of Koh and Shen. As a consequence, we settle down the averaging problems over the quadratic surfaces $V$ in the case when the dimensions $d\geq 4$ are even and $V$ contains a $d/2$-dimensional subspace.
We stduy $L^p-L^r$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, it is proved that if the varieties $V$ in even dimensions have few intersection points with the sphere of zero radius, the same conclusion as in odd dimensional case can be also obtained.
This note establishes sharp $L^p-L^r$ estimates for $X$-ray transforms and Radon transforms in finite fields.
In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout's theorem, we obtain the sufficient and necessary conditions on general curves where sharp $L^p-L^r$ extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao and Iosevich and Koh. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.
We study the number of the vectors determined by two sets in d-dimensional vector spaces over finite fields. We observe that the lower bound of cardinality for the set of vectors can be given in view of an additive energy or the decay of the Fourier transform on given sets. As an application of our observation, we find sufficient conditions on sets where the Falconer distance conjecture for finite fields holds in two dimension. Moreover, we give an alternative proof of the theorem, due to Iosevich and Rudnev, that any Salem set satisfies the Falconer distance conjecture for finite fields.