arXiv subjects
Publications and source records attributed to Doowon Koh.
Let $A$ be a compact set in $\mathbb{R}$, and $E=A^d\subset \mathbb{R}^d$. We know from the Mattila-Sjölin's theorem if $\dim_H(A)>\frac{d+1}{2d}$, then the distance set $Δ(E)$ has non-empty interior. In this paper, we show that the threshold $\frac{d+1}{2d}$ can be improved whenever $d\ge 5$.
Let $f\in \mathbb{R}[x, y, z]$ be a quadratic polynomial that depends on each variable and that does not have the form $g(h(x)+k(y)+l(z))$. Let $A, B, C$ be compact sets in $\mathbb{R}$. Suppose that $\dim_H(A)+\dim_H(B)+\dim_H(C)>2$, then we prove that the image set $f(A, B, C)$ is of positive Lebesgue measure. Our proof is based on a result due to Eswarathasan, Iosevich, and Taylor (Advances in Mathematics, 2011), and a combinatorial argument from the finite field model.
We study both averaging and maximal averaging problems for Product $j$-varieties defined by $Π_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\}$ for $j\in \mathbb F_q^*,$ where $\mathbb F_q^d$ denotes a $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. We prove the sharp $L^p\to L^r$ boundedness of averaging operators associated to Product $j$-varieties. We also obtain the optimal $L^p$ estimate for a maximal averaging operator related to a family of Product $j$-varieties $\{Π_j\}_{j\in \mathbb F_q^*}.$
For $ E\subset \mathbb{F}_q^d$, let $Δ(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset \mathbb{F}_q^d $ are subsets with $|E||F|\gg q^{d+\frac{1}{3}}$ then $|Δ(E)+Δ(F)|> q/2$. They also proved that the threshold $q^{d+\frac{1}{3}}$ is sharp when $|E|=|F|$. In this paper, we provide an improvement of this result in the unbalanced case, which is essentially sharp in odd dimensions. The most important tool in our proofs is an optimal $L^2$ restriction theorem for the sphere of zero radius.
The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide a new $L^2\to L^r$ extension estimate for paraboloids in dimensions $d=4k+3$ and $q\equiv 3\mod 4$, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using \textit{the first association scheme graph} to analyze energy sets, and as a consequence, we obtain new $L^p\to L^4$ extension theorems for spheres of primitive radii in odd dimensions, which break the Stein-Tomas result toward $L^p\to L^4$ which has stood for more than ten years. Most significantly, it follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the $L^p\to L^4$ estimates for spheres with primitive radii are much stronger than those for paraboloids. Based on new estimates, we will also clarify conjectures on finite field extension problem for spheres. This results in a reasonably complete description of finite field extension theorems for spheres. The second purpose is to show that there is a connection between the restriction conjecture associated to paraboloids and the Erdős-Falconer distance conjecture over finite fields. The last is to prove that the Erdős-Falconer distance conjecture holds in odd-dimensional spaces when we study distances between two sets: one set lies on a variety (paraboloids or spheres), and the other set is arbitrary in $\mathbb{F}_q^d$.
In this paper, we solve completely the $L^2\to L^r$ extension conjecture for the zero radius sphere over finite fields. We also obtain the sharp $L^p\to L^4$ extension estimate for non-zero radii spheres over finite fields, which improves significantly the previous result of the first and second listed authors.
In this paper we study $L^p-L^r$ estimates of both extension operators and averaging operators associated with the algebraic variety $S=\{x\in {\mathbb F}_q^d: Q(x)=0\}$ where $Q(x)$ is a nondegenerate quadratic form over the finite field ${\mathbb F}_q.$ In the case when $d\geq 3$ is odd and the surface $S$ contains a $(d-1)/2$-dimensional subspace, we obtain the exponent $r$ where the $L^2-L^r$ extension estimate is sharp. In particular, we give the complete solution to the extension problems related to specific surfaces $S$ in three dimension. In even dimensions $d\geq 2$, we also investigates the sharp $L^2-L^r$ extension estimate. Such results are of the generalized version and extension to higher dimensions for the conical extension problems which Mochenhaupt and Tao studied in three dimensions. The boundedness of averaging operators over the surface $S$ is also studied. In odd dimensions $d\geq 3$ we completely solve the problems for $L^p-L^r$ estimates of averaging operators related to the surface $S.$ On the other hand, in the case when $d\geq 2$ is even and $S$ contains a $d/2$-dimensional subspace, using our optimal $L^2-L^r$ results for extension theorems we, except for endpoints, have the sharp $L^p-L^r$ estimates of the averaging operator over the surface $S$ in even dimensions.
In this paper, we study growth rate of product of sets in the Heisenberg group over finite fields and the complex numbers. More precisely, we will give improvements and extensions of recent results due to Hegyvári and Hennecart (2018).
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.
Let $\mathbb{F}_p$ be a prime field, and ${\mathcal E}$ a set in $\mathbb{F}_p^2$. Let $Δ({\mathcal E})=\{||x-y||: x,y \in {\mathcal E} \}$, the distance set of ${\mathcal E}$. In this paper, we provide a quantitative connection between the distance set $Δ({\mathcal E})$ and the set of rectangles determined by points in ${\mathcal E}$. As a consequence, we obtain a new lower bound on the size of $Δ({\mathcal E})$ when ${\mathcal E}$ is not too large, improving a previous estimate due to Lund and Petridis and establishing an approach that should lead to significant further improvements.
Let $\mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $\det S$ for certain types of subsets $S$ in the ring $M_2(\mathbb F_q)$ of $2\times 2$ matrices with entries in $\mathbb F_q$. For $i\in \mathbb{F}_q$, let $D_i$ be the subset of $M_2(\mathbb F_q)$ defined by $ D_i := \{x\in M_2(\mathbb F_q): \det(x)=i\}.$ Then our results can be stated as follows. First of all, we show that when $E$ and $F$ are subsets of $D_i$ and $D_j$ for some $i, j\in \mathbb{F}_q^*$, respectively, we have $$\det(E+F)=\mathbb F_q,$$ whenever $|E||F|\ge {15}^2q^4$, and then provide a concrete construction to show that our result is sharp. Next, as an application of the first result, we investigate a distribution of the determinants generated by the sum set $(E\cap D_i) + (F\cap D_j),$ when $E, F$ are subsets of the product type, i.e., $U_1\times U_2\subseteq \mathbb F_q^2\times \mathbb F_q^2$ under the identification $ M_2(\mathbb F_q)=\mathbb F_q^2\times \mathbb F_q^2$. Lastly, as an extended version of the first result, we prove that if $E$ is a set in $D_i$ for $i\ne 0$ and $k$ is large enough, then we have \[\det(2kE):=\det(\underbrace{E + \dots + E}_{2k~terms})\supseteq \mathbb{F}_q^*,\] whenever the size of $E$ is close to $q^{\frac{3}{2}}$. Moreover, we show that, in general, the threshold $q^{\frac{3}{2}}$ is best possible. Our main method is based on the discrete Fourier analysis.
In this paper, we study expanding phenomena in the setting of matrix rings. More precisely, we will prove that If $A$ is a set of $M_2(\mathbb{F}_q)$ and $|A|\gg q^{7/2}$, then we have \[|A(A+A)|, ~|A+AA|\gg q^4.\] If $A$ is a set of $SL_2(\mathbb{F}_q)$ and $|A|\gg q^{5/2}$, then we have \[|A(A+A)|, ~|A+AA|\gg q^4.\] We also obtain similar results for the cases of $A(B+C)$ and $A+BC$, where $A, B, C$ are sets in $M_2(\mathbb{F}_q)$.
In this paper we obtain a new lower bound on the Erdős distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb{F}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |Δ(A)| \gg |A|^{\frac{1}{2}+\frac{149}{4214}}.$$ Our result gives a new lower bound of $|Δ{(A)}|$ in the range $|A|\le p^{1+\frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in $\mathbb{F}_p^2$. The latter is the new feature that allows us to improve the previous bound due Stevens and de Zeeuw.
We study the finite field extension estimates for Hamming varieties $H_j, j\in \mathbb F_q^*,$ defined by $H_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\},$ where $\mathbb F_q^d$ denotes the $d$-dimensional vector space over a finite field $\mathbb F_q$ with $q$ elements. We show that although the maximal Fourier decay bound on $H_j$ away from the origin is not good, the Stein-Tomas $L^2\to L^r$ extension estimate for $H_j$ holds.
In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, our previous results. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.
In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be viewed as the Erdős distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that \[\max\{|A+A|, |A^2+A^2|\}\gg |A|^{1+\frac{1}{42}}, ~|A+A^2|\gg |A|^{1+\frac{1}{84}}.\]
Let $\mathbb{F}_p$ be a prime field of order $p>2$, and $A$ be a set in $\mathbb{F}_p$ with very small size in terms of $p$. In this note, we show that the number of distinct cubic distances determined by points in $A\times A$ satisfies \[|(A-A)^3+(A-A)^3|\gg |A|^{8/7},\] which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that \[\max \left\lbrace |A+A|, |f(A, A)|\right\rbrace\gg |A|^{6/5},\] where $f(x, y)$ is a quadratic polynomial in $\mathbb{F}_p[x, y]$ that is not of the form $g(αx+βy)$ for some univariate polynomial $g$.
In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in $SL_2(\mathbb{F}_p)$ with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyvári and Hennecart.