arXiv subjects

Daewoong Cheong

Publications and source records attributed to Daewoong Cheong.

17 recordsLinked to original sources

arXiv2026

The Mattila-Sj\"olin problem for the k-distance over a finite field

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.

math.CO↗

arXiv2025

Additive structures imply more distances in $\mathbb{F}_q^d$

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.

math.CO↗

arXiv2025

A functional Loomis-Whitney type inequality in the Heisenberg group and projection theorems over finite fields

We establish functional Loomis--Whitney type inequalities in the finite Heisenberg group $\mathbb{H}^n(\mathbb{F}_q)$. For $n=1$, we determine the sharp region of exponents $(u_1,u_2)$ for which the Heisenberg Loomis--Whitney inequality \[ \frac{1}{q^3}\sum_{(x,t)\in \mathbb{H}^1(\mathbb{F}_q)} f_1(\pi_1(x,t))\,f_2(\pi_2(x,t)) \;\lesssim\; \|f_1\|_{L^{u_1}(\mathbb{F}_q^2,dx)}\|f_2\|_{L^{u_2}(\mathbb{F}_q^2,dx)} \] holds uniformly in $q$, namely \[ \frac{1}{u_1}+\frac{2}{u_2}\le 2 \quad\text{and}\quad \frac{2}{u_1}+\frac{1}{u_2}\le 2, \] which includes the endpoint estimate $L^{\frac{3}{2}}\times L^{\frac{3}{2}}\to L^1$. For general $n$, we prove the symmetric multilinear estimate at the endpoint exponent $ u=\frac{n(2n+1)}{n+1}, $ using an induction on $n$ that exploits the Heisenberg fiber structure together with a multilinear interpolation scheme. Specializing to indicator functions yields a sharp Loomis--Whitney type set inequality bounding $|K|$ for every finite $K\subset \mathbb{H}^n(\mathbb{F}_q)$ in terms of the sizes of its $2n$ Heisenberg projections $\{\pi_j(K)\}_{j=1}^{2n}$, and in particular, \[ \max_{1\le j\le 2n} |\pi_j(K)| \;\gtrsim_n\; |K|^{\frac{2n+1}{2(n+1)}}\,q^{-\frac{1}{2(n+1)}}. \] This result is optimal up to absolute constants. Moreover, when $n=1$ and $|K|>q$, we obtain a stronger statement via Vinh's point--line incidence theorem. We also discuss connections to a boundedness problem for multilinear forms/operators over finite fields studied by Bhowmik, Iosevich, Koh, and Pham (2025), and to orthogonal projection/covering questions in $\mathbb{F}_q^{2n+1}$ studied by Chen (2018).

math.CO↗

arXiv2023

Counting maximal isotropic subbundles of orthogonal bundles over a curve

Let $C$ be a smooth projective curve and $V$ an orthogonal bundle over $C$. Let $\IQeV$ be the isotropic Quot scheme parameterizing degree $e$ isotropic subsheaves of maximal rank in $V$. We give a closed formula for intersection numbers on components of $\IQeV$ whose generic element is saturated. As a special case, for $g \ge 2$, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in \cite{Ho}, and of the symplectic case studied in \cite{CCH1}.

math.AG↗

arXiv2021

Mattila--Sjölin type functions: A finite field model

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 $\mathbb{F}_q^d$.

math.CA↗

arXiv2020

An asymmetric bound for sum of distance sets

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.

math.NT↗

arXiv2020

Irreducibility of Lagrangian Quot schemes over an algebraic curve

Let $C$ be a complex projective smooth curve and $W$ a symplectic vector bundle of rank $2n$ over $C$. The Lagrangian Quot scheme $LQ_{-e}(W)$ parameterizes subsheaves of rank $n$ and degree $-e$ which are isotropic with respect to the symplectic form. We prove that $LQ_{-e}(W)$ is irreducible and generically smooth of the expected dimension for all large $e$, and that a generic element is saturated and stable.

math.AG↗

arXiv2020

Isotropic Quot schemes of orthogonal bundles over a curve

We study the isotropic Quot schemes $IQ_e (V)$ parameterizing degree $e$ isotropic subsheaves of maximal rank of an orthogonal bundle $V$ over a curve. The scheme $IQ_e (V)$ contains a compactification of the space $IQ^o_e (V)$ of degree $e$ maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [6]. We observe that for certain topological types of $V$, the scheme $IQ_e (V)$ is empty for all $e$. In the remaining cases, for infinitely many $e$ there are irreducible components of $IQ_e (V)$ consisting entirely of nonsaturated subsheaves, and so $IQ_e (V)$ is strictly larger than the closure of $IQ^o_e (V)$. As our main result, we prove that for any orthogonal bundle $V$ and for $e \ll 0$, the closure $\overline{IQ^o_e (V)}$ of $IQ^o_e (V)$ is either empty or consists of one or two irreducible connected components, depending on $°(V)$ and $e$. In so doing, we also characterize the nonsaturated part of $\overline{IQ^o_e (V)}$ when $V$ has even rank.

math.AG↗

arXiv2019

Galkin's Lower bound Conjecure for Lagrangian and orthogonal Grassmannians

Let $M$ be a Fano manifold, and $H^\star(M;\mathbb{C})$ be the quantum cohomology ring of $M$ with the quantum product $\star.$ For $σ\in H^*(M;\mathbb{C})$, denote by $[σ]$ the quantum multiplication operator $σ\star$ on $H^*(M;\mathbb{C})$. It was conjectured several years ago \cite{GGI, GI} and has been proved for many Fano manifols \cite{CL1, CH2, LiMiSh, Ke}, including our cases, that the operator $[c_1(M)]$ has a real valued eigenvalue $δ_0$ which is maximal among eigenvaules of $[c_1(M)]$. Galkin's lower bound conjecture \cite{Ga} states that for a Fano manifold $M,$ $δ_0\geq \mathrm{dim} \ M +1,$ and the equlity holds if and only if $M$ is the projective space $\mathbb{P}^n.$ In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.

math.AG↗

arXiv2019

Distribution of determinant of sum of matrices

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.

math.CO↗

arXiv2019

Counting maximal Lagrangian subbundles over an algebraic curve

Let $C$ be a smooth projective curve and $W$ a symplectic bundle over $C$. Let $LQ_e (W)$ be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves $E \subset W$ of degree $e$. We give a closed formula for intersection numbers on $LQ_e (W)$. As a special case, for $g \ge 2$, we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [13].

math.AG↗

arXiv2019

Extension theorems for Hamming varieties over finite fields

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.

math.CA↗

arXiv2017

Quantum multiplication operators for Lagrangian and orthogonal Grassmannians

In this article, we make a close analysis on quantum multiplication operators on the quantum cohomology rings of Lagrangian and orthogonal Grassmannians, and give an explicit description on all simultaneous eigenvectors and the corresponding eigenvalues for these operators. As a result, we show that Conjecture $\mathcal{O}$ of Galkin, Golyshev and Iritani holds for these manifolds.

math.AG↗

arXiv2016

On the Conjecture $\mathcal{O}$ of GGI for $G/P$

In this paper, we show that general homogeneous manifolds $G/P$ satisfy Conjecture $\mathcal{O}$ of Galkin, Golyshev and Iritani which `underlies' Gamma conjectures I and II of them. Our main tools are the quantum Chevalley formula for $G/P$ and a theory on nonnegative matrices including Perron-Frobenius theorem.

math.AG↗

arXiv2016

Equivariant compactifications of a nilpotent group by $G/P$

Let $G$ be a simple complex algebraic group, $P$ a parabolic subgroup of $G$ and $N$ the unipotent radical of $P.$ The so-called equivariant compactification of $N$ by $G/P$ is given by an action of $N$ on $G/P$ with a dense open orbit isomorphic to $N$. In this article, we investigate how many such equivariant compactifications there exist. Our result says that there is a unique equivariant compactification of $N$ by $G/P$, up to isomorphism, except $¶^n$.

math.AG↗

arXiv2015

Orbifold Quasimap Theory

We extend to orbifolds the quasimap theory of arXiv:0908.4446 and arXiv:1106.3724, as well as the genus zero wall-crossing results from arXiv:1304.7056 and arXiv:1401.7417. As a consequence, we obtain generalizations of orbifold mirror theorems, in particular, of the mirror theorem for toric orbifolds recently proved independently by Coates, Corti, Iritani, and Tseng (arXiv:1310.4163).

math.AG↗

arXiv2007

Quantum Cohomology Rings of Lagrangian and Orthogonal Grassmannians and Total Positivity

We verify in an elementary way a result of Peterson for the maximal orthogonal and Lagrangian Grassmannians, and then find Vafa-Intriligator type formulas which compute their 3-point, genus zero Gromov-Witten invariants. Finally we study total positivity of the related Peterson varieties and show that Rietsch's conjecture about the total positivity holds for these cases.

math.QA↗