arXiv subjects

Dung The Tran

Publications and source records attributed to Dung The Tran.

12 recordsLinked to original sources

arXiv2026

Sharp refined-direction Kakeya estimates in finite Heisenberg groups

Let $n\geq 2$ and let $q$ be an odd prime power. The first aim of this paper is to prove that, for every $E\subset \mathbb{H}_n(\mathbb{F}_q)$ and every $λ>0$, the following sharp rich-direction estimate holds \[ \left| \left\{ \vartheta\in D_n: M^{\mathrm{rd}}_{\mathbb{H}_n}\mathbf{1}_E(\vartheta)\geqλ \right\} \right| \lesssim_n q^{2n-1}|E|λ^{-2n}. \] The second aim is to determine, for every $1\leq u,v\leq\infty$, the sharp exponent of $q$ in the corresponding $\ell^u\to\ell^v$ estimate. More precisely, we prove that \[ A_n^{\mathrm{rd}}(u,v) = \max\left\{ \frac{2n-1}{v},\ 1-\frac1u,\ \frac{2n}{v}-\frac1u,\ 1+\frac{2n}{v}-\frac{2n+1}{u} \right\}. \] The proof combines the polynomial method with multiplicities and a probabilistic covering argument based on the action of the affine symplectic group.

math.CA↗

arXiv2026

Erdős--Falconer distance conjecture from an analytic perspective

Let $q$ be an odd prime power and let $V=\F_q^{2m}$, equipped with $Q(x)=x_1^2+\cdots+x_{2m}^2$. We develop a semidefinite Delsarte framework for the two-set Erdős--Falconer distance problem over $V$. The framework reduces the natural $q^m$-scale positive-proportion theorem to a uniform $L^1$ anti-concentration statement for positive convex combinations of classical Kloosterman sums. Assuming this Kloosterman anti-concentration conjecture, we prove that, for every $0<α<\frac12$, there exists a constant $C_{m,α}$ such that \[ \min\{|E|,|F|\}\ge C_{m,α}q^m \quad\Longrightarrow\quad |Δ_Q^\times(E,F)|>α(q-1) \] for all $E,F\subset\F_q^{2m}$. More generally, a $q^{-θ}$-level version of the Kloosterman input yields the geometric threshold $q^{m+θ}$. Unconditionally, we establish a uniform $q^{-\frac13}$-level $L^1$ lower bound whenever the support has size at most $α(q-1)$, with $0<α<\frac12$. Consequently, \[ \min\{|E|,|F|\}\ge C_{m,α}q^{m+\frac13} \quad\Longrightarrow\quad |Δ_Q^\times(E,F)|>α(q-1). \] The proof uses positive semidefinite $2\times2$ Gram matrices on quadratic frequency shells, the shell Fourier transform in even dimension, and a minimax separation argument that produces a uniform signed combination of Kloosterman columns. We also provide evidence for the constant-scale Kloosterman conjecture and discuss its limitations near full support.

math.CO↗

arXiv2026

Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications

Let $q$ be an odd prime power. We study Kakeya maximal operators associated with horizontal lines in the finite Heisenberg groups $\mathbb H_n(\mathbb F_q)$. Our principal object is the refined-direction maximal operator, whose parameter records the projective horizontal direction together with the central homogeneous coordinate determined by horizontality. In rank one, we prove \[ \|M_{\mathbb H_1}^{\mathrm{rd}}F\|_{\ell^2(\mathcal D_1)} \lesssim q^{\frac{1}{2}} \|F\|_{\ell^2(\mathbb H_1(\mathbb F_q))}, \] where the exponent $\frac{1}{2}$ is sharp. Combining this estimate with endpoint bounds and interpolation, we determine the exact mixed-norm growth exponent: \[ A^{\mathrm{rd}}_1(u,v) = \max\left\{ \frac1v,\, 1-\frac1u,\, \frac2v-\frac1u,\, 1+\frac2v-\frac3u \right\}, \qquad 1\le u,v\le\infty. \] As a consequence, if $E\subset\mathbb H_1(\mathbb F_q)$ meets, in at least $m$ points, a horizontal line in each refined direction from $Ω\subset D_1$, then \[ |E|\gtrsim \frac{m^2|Ω|}{q}. \] As a benchmark, we also analyze the coarser operator parameterized only by projective horizontal directions and determine its exact $\ell^u\to\ell^v$ growth exponent in every rank. In rank one, this benchmark is established by a self-contained $TT^*$ argument rather than polynomial vanishing, and the same planar estimate reappears as the zero-central-frequency component of the refined-direction proof. The nonzero central frequencies are controlled by Plancherel, character orthogonality, and a bounded-fiber property of an explicit quadratic map. Thus, the sharp refined-direction estimate is obtained by purely Fourier-analytic methods.

math.CO↗

arXiv2026

A sharp point-sphere incidence bound for $(u, s)$-Salem sets

We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of $(4,s)$-Salem sets, which quantify pseudorandomness via fourth-order additive energy, we prove that if $P\subset \mathbb{F}_q^d$ is a $(4,s)$-Salem set with $s\in \big( \frac{1}{4}, \frac{1}{2} \big]$ and $|P|\ll q^{ \frac{d}{4s}}$, then for any finite family $S$ of spheres in $\mathbb{F}_q^d$, \[ \bigg| I(P,S)-\frac{|P||S| }{q} \bigg| \ll q^{\frac{d}{4}}\,|P|^{1-s}\,|S|^{\frac{3}{4}}. \] This estimate improves the classical point-sphere incidence bounds for arbitrary point sets across a broad parameter range. The proof combines additive energy estimates with a lifting argument that converts point-sphere incidences into point-hyperplane incidences in one higher dimension while preserving the $(4,s)$-Salem property. As applications, we derive refined bounds for unit distances and sum-product type phenomena, and we extend the method to $(u,s)$-Salem sets for even moments $u\ge4$.

math.CO↗

arXiv2026

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 $Δ(E) := \{\|\mathbf{x} - \mathbf{y}\| : \mathbf{x}, \mathbf{y} \in E\}$, where $\|\cdot\|$ denotes the standard quadratic form. We investigate the Erdős--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 $Λ_4(E)$, we prove that quantitative gains in $Λ_4(E)$ 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↗

arXiv2026

Raimi's theorem for manifolds with circle symmetry

Raimi's classical theorem establishes a partition of the natural numbers with a remarkable unavoidability property: for every finite coloring of $\mathbb{N}$, there is a color class whose translate meets both parts of the partition in infinitely many points. Recently, Kang, Koh, and Tran have extended this phenomenon to the circle group, proving that there exists a measurable partition of the circle such that every finite measurable cover admits a rotation whose image meets each part of the partition in positive measure. This paper shows that this phenomenon extends beyond compact abelian groups to a wide class of non-group geometric surfaces that still exhibit \textit{a hidden one-dimensional symmetry}. Specifically, we establish analogs of Raimi's theorem for three families of surfaces (with their natural surface measures): the unit sphere $S^{n-1} \subset \mathbb{R}^n$, rotational power surfaces (such as cones and paraboloids), and circular cylindrical surfaces. The common feature is that each of these surfaces carries a natural measure-preserving action of the circle group by rotation in a fixed plane and admits a measurable trivialization as a product $C \times Y$. This circle-bundle structure allows the measurable Raimi partition on the base circle to be lifted to an unavoidable partition on the manifold. Our approach is unified through a general circle-bundle theorem, which reduces all three geometric cases to verifying suitable equivariance and product disintegration properties of the surface measure.

math.CO↗

arXiv2026

Parabolic distance in $\mathbb F_q^2$: a sharp exponent and new results

We study the parabolic variant of the Erd\H os--Falconer distance problem in finite fields. That is, if $q$ is odd, we seek size thresholds beyond which any subset $E\subset \mathbb F_q^2$ will determine many distinct parabolic distances. This problem has a rich history because the parabolic distance functional shares many properties with the standard distance functional, but exhibits many distinct behaviors. Here we begin with rather standard Fourier analytic arguments, but diverge into additive combinatorics to handle the central obstructions. We provide a suite of positive results and corresponding sharpness examples.

math.CO↗

arXiv2026

Existence of Quantum Splines via Fourth-Order Gradient Flows

We establish a rigorous existence theory for the quantum splines introduced by Brody, Holm, and Meier in Physical Review Letters (2012). These curves arise as solutions of a variational problem on the unitary group describing optimally controlled quantum evolutions. By formulating the problem within a geometric gradient flow framework for Riemannian spline interpolation, we construct a well-posed fourth order evolution whose asymptotic limits realize the desired quantum splines. The analysis requires adapting the variational structure to boundary conditions dictated by the physical model, which are not directly amenable to the setting in our recently developed framework for gradient flows of Riemannian spline interpolation. We show that, despite these difficulties, the modified system admits a rigorous analytical treatment, yielding both existence and a constructive procedure for generating quantum splines. Our results provide a mathematical foundation for the variational description of smooth quantum control trajectories and clarify the analytical structure underlying their formation.

math.OC↗

arXiv2026

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 \[ \frac{1}{q^3}\sum_{(x,t)\in \mathbb{H}^1(\mathbb{F}_q)} f_1(π_1(x,t))\,f_2(π_2(x,t)) \;\lesssim\; \|f_1\|_{L^{u_1}(\mathbb{F}_q^2,dx)}\|f_2\|_{L^{u_2}(\mathbb{F}_q^2,dx)} \] 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 $\{π_j(K)\}_{j=1}^{2n}$, and in particular, \[ \max_{1\le j\le 2n} |π_j(K)| \;\gtrsim_n\; |K|^{\frac{2n+1}{2(n+1)}}\,q^{-\frac{1}{2(n+1)}}. \] 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↗

arXiv2025

Raimi's theorem for the $n$-dimensional torus

We extend Raimi's classical partition theorem to the continuous setting of the circle and $n$-dimensional torus. Building on recent work of Hegyvári, 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.

math.CO↗

arXiv2025

Higher-order Riemannian spline interpolation problems: a unified approach by gradient flows

This paper addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks, providing a novel framework for tackling these challenges. This method notably extends to the variational spline interpolation problem on Lie groups, which is frequently encountered in mechanical optimal control theory. As a result, our work contributes to both geometric control theory and statistical shape data analysis. We rigorously prove the existence of global solutions in Hölder spaces for the gradient flow and demonstrate that the asymptotic limits of these solutions validate the existence of solutions to the variational spline interpolation problem. This constructive proof also offers insights into potential numerical schemes for finding such solutions, reinforcing the practical applicability of our approach.

math.OC↗

arXiv2022

The Erdos-Falconer distance problem in the tree setting

The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set $Δ(A)$ has positive Lebesgue measure. In a very recent paper, Murphy, Petridis, Pham, Rudnev, and Stevens (2022) proved the prime field version of this result, namely, for $E\subset\mathbb{F}_p^2$ with $|E|\gg p^{5/4}$, there exist many points $x\in E$ such that the number of distinct distances from $x$ is at least $cp$. The main purpose of this paper is to provide extensions in a very general structure of pinned trees, which is inspired by the recent work due to Ou and Taylor (2021).

math.CO↗