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Doowon Koh

Publications and source records attributed to Doowon Koh.

63 records · Page 4Linked to original sources
arXiv2010

Harmonic analysis related to homogeneous varieties in three dimensional vector space over finite fields

In this paper we study extension problems, averaging problems, and generalized Erdos-Falconer distance problems associated with arbitrary homogeneous varieties in three dimensional vector space over finite fields. In the case when homogeneous varieties in three dimension do not contain any plane passing through the origin, we obtain the general best possible results on aforementioned three problems. In particular, our results on extension problems recover and generalize the work due to Mockenhaupt and Tao who completed the particular conical extension problems in three dimension. Investigating the Fourier decay on homogeneous varieties, we give the complete mapping properties of averaging operators over homogeneous varieties in three dimension. In addition, studying the generalized Erd\H os-Falconer distance problems related to homogeneous varieties in three dimensions, we improve the cardinality condition on sets where the size of distance sets is nontrivial. Finally, we address a question of our problems for homogeneous varieties in higher odd dimensions.

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arXiv2010

The generalized Erdos-Falconer distance problems in vector spaces over finite fields

In this paper we study the generalized Erdos-Falconer distance problems in the finite field setting. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular, we develop a simple formula for estimating the cardinality of distance sets determined by diagonal polynomials. As a result, we generalize the spherical distance problems due to Iosevich and Rudnev and the cubic distance problems due to Iosevich and Koh. Moreover, our results are of higher dimensional version for Vu's work on two dimension. In addition, we set up and study the generalized pinned distance problems in finite fields. We give a nice generalization of some recent work in which the pinned distance problems related to spherical distances were investigated. Discrete Fourier analysis and exponential sum estimates play an important role in our proof.

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arXiv2009

Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $α>0$ such that $|Δ(E)| \gtrsim q$ whenever $|E| \gtrsim q^α$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $Δ(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. In two dimensions we improve the known exponent to $\tfrac{4}{3}$, consistent with the corresponding exponent in Euclidean space obtained by Wolff. The pinned distance set $Δ_y(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x\in E\}$ for a pin $y\in E$ has been studied in the Euclidean setting. Peres and Schlag showed that if the Hausdorff dimension of a set $E$ is greater than $\tfrac{d+1}{2}$ then the Lebesgue measure of $Δ_y(E)$ is positive for almost every pin $y$. In this paper we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set $Π_y(E)=\{x\cdot y: x\in E\}$. Under the additional assumption that the set $E$ has cartesian product structure we improve the pinned threshold for both distances and dot products to $\frac{d^2}{2d-1}$. A generalization of the Falconer distance problem is determine the minimal $α>0$ such that $E$ contains a congruent copy of every $k$ dimensional simplex whenever $|E| \gtrsim q^α$. Here the authors improve on known results (for $k>3$) using Fourier analytic methods, showing that $α$ may be taken to be $\frac{d+k}{2}$.

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arXiv2008

Extension Theorems for Paraboloids in the Finite Field Setting

In this paper we study the $L^p-L^r$ boundedness of the extension operators associated with paraboloids in vector spaces over finite fields.In higher even dimensions, we estimate the number of additive quadruples in the subset $E$ of the paraboloids, that is the number of quadruples $(x,y,z,w) \in E^4$ with $x+y=z+w.$ As a result, in higher even dimensions, we improve upon the standard Tomas-Stein exponents which Mockenhaupt and Tao obtained for the boundedness of extension operators for paraboloids by estimating the decay of the Fourier transform of measures on paraboloids. In particular, we obtain the sharp $L^p-L^4$ bound up to endpoints in higher even dimensions. Moreover, we also study the $L^2-L^r$ estimates.In the case when -1 is not a square number in the underlying finite field, we also study the $L^p-L^r$ bound in higher odd dimensions.The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.

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arXiv2008

Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields

We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas-Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimates for Gauss and Kloosterman sums and their variants play an important role.

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arXiv2008

Distance graphs in vector spaces over finite fields, coloring and pseudo-randomness

In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance graphs and show that sufficiently large subsets of $d$-dimensional vector spaces over finite fields contain every possible finite configurations.

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arXiv2008

Generalized incidence theorems, homogeneous forms, and sum-product estimates in finite fields

In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in \cite{HI07} and \cite{HIKR07}. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions. In particular, our machinery enables us to prove that if $E \subset {\Bbb F}_q^d$, $d \ge 4$, the $d$-dimensional vector space over a finite field ${\Bbb F}_q$, of size much greater than $q^{\frac{d}{2}}$, and if $E$ is a product set, then the set of volumes of $d$-dimensional parallelepipeds determined by $E$ covers ${\Bbb F}_q$. This result is sharp as can be seen by taking $E$ to equal to $A \times A \times ... \times A$, where $A$ is a sub-field of ${\Bbb F}_q$ of size $\sqrt{q}$. In three dimensions we establish the same result if $|E| \gtrsim q^{15/8}$. We prove in three dimensions that the set of volumes covers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^{3/2}$. Finally we show that in three dimensions the set of volumes covers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^2$, without any further assumptions on $E$, which is again sharp as taking $E$ to be a 2-plane through the origin shows.

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arXiv2007

Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture

We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $\mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.

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