arXiv subjects

Thang Pham

Publications and source records attributed to Thang Pham.

At least 19 recordsLinked to original sources

arXiv2026

Congruence Classes in $\mathbb{F}_p^2$: Sharp Results via an Energy Approach

Let $T_k(E)$ denote the set of congruence classes of ordered $k$-tuples of pairwise distinct points of $E$. Let $p\equiv3\pmod4$ be prime. For $E\subset\mathbb{F}_p^2$ with $3\leq|E|\leq p^{3/4}$, we prove that $|T_3(E)|\gg|E|^{11/6}$; for $4\leq|E|\leq p^{3/4}$, we prove that $|T_4(E)|\gg|E|^3/\log|E|$. For every fixed $k\geq5$ and $k\leq|E|\leq p^{3(k-2)/(3k-4)}$, we prove that $|T_k(E)|\gg_k|E|^{k-1}$. The proofs proceed by bounding the moments of the overlap function of rigid motions. The main inputs are an exact identity involving the distance energy and an incidence bound obtained by viewing rigid motions as lines over $\mathbb{F}_p(i)$. The bound for $k=4$ is sharp up to a logarithmic factor, while the bounds for $k\geq5$ are sharp up to constants.

math.CO↗

arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq \, 34/11$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erdős--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗

arXiv2026

Subset-Sum Density Realization in Locally Finite Abelian Groups

Let $G$ be a countable locally finite abelian group, and let \[ G_1\leq G_2\leq\cdots, \qquad \bigcup_{i\geq1}G_i=G, \] be any filtration of $G$ by finite subgroups. For $A\subseteq G$, let $\mathcal P(A)$ denote the set of all finite subset sums of elements of $A$, and let $2G:=\{2g:g\in G\}$. We prove that $|2G|=\infty$ if and only if for every filtration and every interval $[α,β]\subseteq[0,1]$, there exists $A\subseteq G$ such that the set of limit points of \[ \left( \frac{|\mathcal P(A)\cap G_i|}{|G_i|} \right)_{i\geq1} \] is exactly $[α,β]$. We also prove a positive-density result that does not require $|2G|=\infty$: if $G$ is infinite and $\mathcal P(A)$ has positive upper density along the given filtration, then there is an infinite set $B\subseteq \mathcal P(A)$ such that $B+B\subseteq \mathcal P(A)$. Combining this with the realization theorem, we show that, when $|2G|=\infty$, every interval $[α,β]$ with $0\leqα\leqβ\leq1$ and $β>0$ can be realized by a set $A$ with this additional property.

math.CO↗

arXiv2026

Entropy Expansion for General Polynomial Images of Frostman Random Variables

We prove dyadic entropy expansion for the observables $X+Y$ and $f(X,Y)$ under Frostman nonconcentration hypotheses on a prescribed, possibly dependent law. For an integer $n\ge1$, write $H_n(Z)=H(\lfloor 2^nZ\rfloor)$ for the base-two Shannon entropy at resolution $2^{-n}$; thus $n$ indexes the fineness of the dyadic discretization. For every $0 4/3$, we obtain $\max\{H_n(X+Y),H_n(f(X,Y))\}\ge (\frac{s_1+s_2}{2}+\varepsilon)n-O(1)$ for an explicit $\varepsilon>0$. Here the baseline averages the two Frostman exponents. We also classify the exceptional coordinate representations and construct obstructions for algebraic affine directions with Frostman constants uniform in the scale.

math.CA↗

arXiv2026

Materials for Quantum Information Science: Roles in the Quantum Evolution 2.0

Quantum information science is entering a second phase, the Quantum Evolution 2.0, in which the challenge has shifted from demonstrating coherent control of individual quantum states to building scalable multi-qubit processors and networks. This transition places materials science at the center of the field. Across superconducting circuits, quantum defects, quantum photonic devices, and emerging materials platforms, including two-dimensional materials and heterostructures, performance is now limited less by device design than by poorly controlled surfaces, buried interfaces, and defects whose atomic identities remain incompletely known. This review surveys the materials challenges of these quantum platforms together with the characterization methods needed to resolve them. For each platform we identify the dominant decoherence mechanisms, the current state of materials understanding, and the most pressing open materials problems. A cross-platform comparison then reveals a shared structure-coherence problem. The implicated material chemistry recurs across platforms, involving light elements in disordered or buried environments, yet no platform can quantitatively connect a specific atomic-scale structure to a measured change in coherence. We close by identifying three needs, mechanistic understanding of decoherence at the atomistic level, high-throughput proxy metrics predictive of device performance, and characterization tools built for quantum materials, whose resolution would advance coherence, scalability, and integration across all platforms.

quant-ph↗

arXiv2026

On $L^2$ estimates for quadratic images of product Frostman measures

Let $f\in\mathbb R[x,y,z]$ be a fixed non-degenerate quadratic polynomial. Given an $α$-Frostman probability measure $μ$ supported on $[0,1]$ with $α\in(0,1)$, consider the pushforward measure $ν=f_{\#}(μ\timesμ\timesμ)$ on $\mathbb R$. We prove the following $L^2$ energy estimate: for a fixed nonnegative Schwartz function $φ$ with $\intφ=1$ and $φ_δ(t)=δ^{-1}φ(t/δ)$, there exist $ε>0$ and $δ_{0}>0$ (depending only on $α$ and the coefficients of $f$) such that \[ \int_{\mathbb R}(φ_δ*ν(t))^{2}\,dt \ \lesssim\ δ^{α+ε-1} \qquad \text{for all } δ\in(0,δ_{0}]. \] The proof expands the $L^2$ energy into a weighted six-fold coincidence integral and reduces the main contribution to a planar incidence problem after a controlled change of variables. The key new input is an incidence estimate for point sets that arise as bi-Lipschitz images of a Cartesian product $M\times M$ of a $δ$-separated and non-concentrated set $M$, yielding a power saving beyond what is available from separation and non-concentration alone. We also give examples showing that bounded support and Frostman-type hypotheses are necessary for such $L^{2}$ control.

math.CA↗

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

Distribution of simplices in the discrete and continuous settings

In this paper, we study the distribution of simplices in both discrete and continuous settings. Let $q$ be an odd prime power, let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^d$, and let $2\leq k\leq d-1$. We prove that every set $E\subset\mathbb F_q^d$ with \[ |E|\geq C_{d,k}q^{β_{d,k}}, \qquad β_{d,k}= \begin{cases} \displaystyle \frac{d+k}{2}-\frac{k-1}{k+1}, & d-k\ \text{even},\\[2mm] \displaystyle \frac{d+k-1}{2}, & d-k\ \text{odd}, \end{cases} \] determines a positive proportion of all ordered nondegenerate $k$-simplex congruence classes. This improves the previous exponent due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2017), and is sharp when $d-k$ is odd. In the Euclidean setting, we prove that if $E\subset\mathbb R^d$ is compact and $\dim_{\mathrm H}(E)>d-1$, then there exists a Frostman probability measure $μ$, supported on $E$, and a set of pins of full $μ$-measure such that the pinned distance configuration measure for labeled $(d-1)$-simplices is absolutely continuous at every such pin. We also show that the same conclusion holds when $E\subset\mathbb R^d$ is a compact Salem set with $\dim_{\mathrm H}(E)>k$.

math.NT↗

arXiv2026

On Erdos-Falconer distance problem in even dimensions

Let $q$ be an odd prime power and $\mathbb{F}_q$ be the finite field of order $q$. We prove an extraction theorem for the Erdős-Falconer distance conjecture in even dimensions, showing that the conjecture for all even dimensions reduces to the planar case. As consequences, we obtain improved thresholds on the pinned distance problem and the distribution of triangles, achieving new records of $\frac{d}{2}+\frac{1}{4}$ over prime fields and $\frac{d+1}{2}+\frac{1}{10}$ over arbitrary finite fields, respectively.

math.NT↗

arXiv2026

On the distance problem over finite p-adic rings

In this paper, we study the distance problem in the setting of finite p-adic rings. In odd dimensions, our results are essentially sharp. In even dimensions, we clarify the conjecture and provide examples to support it. Surprisingly, compared to the finite field case, in this setting, we are able to provide a large family of sets such that the distance conjecture holds. By developing new restriction type estimates associated to circles and orbits, with a group theoretic argument, we will prove the $4/3$-parallel result in the two dimensions. This answers a question raised by Alex Iosevich. In a more general scenario, the existence/distribution of geometric/graph configurations will be also considered in this paper. Our results present improvements and extensions of recent results due to Ben Lichtin (2019, 2023). In comparison with Lichtin's method, our approach is much simpler and flexible, which is also one of the novelties in this paper.

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

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

On Fourier decay and the distance set problem

We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild assumptions, we are able to beat the $d/2$ dimension threshold in dimensions $d \geq 5$. For example, we show that (in any ambient spatial dimension $d$) a Borel set with Fourier dimension at least $2$ has a distance set of full Hausdorff dimension. We also show that (in any ambient spatial dimension $d$) a Borel set with Fourier spectrum at least $d/4+1$ at $θ=1/2$ has a distance set of full Hausdorff dimension. In particular, this can hold for sets with Fourier dimension zero (provided $d \geq 4$). We also consider pinned variants of these problems and construct examples that demonstrate the sharpness (or near sharpness) of our results.

math.CA↗

arXiv2026

Vapor-liquid-solid growth of unconventional nanowires

Vapor liquid solid (VLS) growth is one of the most widely used routes for nanowire synthesis. For conventional semiconductor nanowires, here we refer to group IV and III-V systems, decades of work have established VLS growth across diverse vapor-phase methods and enabled substantial control over morphology, crystal phase, and structural modulation. In contrast, comparable deterministic control has not yet been achieved for many non-conventional nanowire classes, including oxides, carbides, and chalcogenides, despite their predicted functional properties and broad application potential. Here we survey and categorize the literature on VLS and VLS-related synthesis of these non-conventional nanowires, highlighting key similarities and differences relative to the group IV and III-V baseline. We analyze mechanistic and potential factors that underlie the lag in synthesis development, including constraints associated with precursor's chemistry and delivery, seed particle composition and dynamics, and competing non-catalytic nucleation and growth pathways. The review is grouped into three main sections, according to the order in which each step takes place during a nanowire growth process, namely precursor delivery, seed particle formation, and nucleation and growth. Each section starts with a brief discussion of what has been achieved in group IV and III-V nanowires as a baseline, followed by similar as well as unique aspects in other material classes. Each section concludes with challenges and opportunities, where we discuss how insights developed in one nanowire system can inform progress in others, ultimately paving the way for more deterministic synthesis and integration of complex one-dimensional nanomaterials.

cond-mat.mtrl-sci↗

arXiv2026

Toward Reliable Evaluation of LLM-Based Financial Multi-Agent Systems: Taxonomy, Coordination Primacy, and Cost Awareness

Multi-agent systems based on large language models (LLMs) for financial trading have grown rapidly since 2023, yet the field lacks a shared framework for understanding what drives performance or for evaluating claims credibly. This survey makes three contributions. First, we introduce a four-dimensional taxonomy, covering architecture pattern, coordination mechanism, memory architecture, and tool integration; applied to 12 multi-agent systems and two single-agent baselines. Second, we formulate the Coordination Primacy Hypothesis (CPH): inter-agent coordination protocol design is a primary driver of trading decision quality, often exerting greater influence than model scaling. CPH is presented as a falsifiable research hypothesis supported by tiered structural evidence rather than as an empirically validated conclusion; its definitive validation requires evaluation infrastructure that does not yet exist in the field. Third, we document five pervasive evaluation failures (look-ahead bias, survivorship bias, backtesting overfitting, transaction cost neglect, and regime-shift blindness) and show that these can reverse the sign of reported returns. Building on the CPH and the evaluation critique, we introduce the Coordination Breakeven Spread (CBS), a metric for determining whether multi-agent coordination adds genuine value net of transaction costs, and propose minimum evaluation standards as prerequisites for validating the CPH.

cs.MA↗

arXiv2026

Packing sets under finite groups via algebraic incidence structures

Let $G$ be a finite group acting on a vector space $V = \mathbb{F}_p^n$ over a prime field. Given finite sets $S \subset G$ and $E \subset V$, we study the restricted orbit union $S(E) = \bigcup_{g\in S} g(E)$ and establish quantitative lower bounds for $|S(E)|$ in terms of $|S|$, $|E|$, and natural structural conditions. This finite field packing problem has connections to distance geometry, configuration counting, and expanding graphs. For $G = SL_2(\mathbb{F}_p)$ acting on $\mathbb{F}_p^2$, we prove that $$|S(E)| \gg \min\left\lbrace p^2, \frac{|S||E|}{p^2}\right\rbrace,$$ which is sharp. Under geometric non-concentration conditions on $E$ and subgroup-avoidance hypotheses on $S$, we obtain a power-saving improvement of the form $$|S(E)|\gg \min \left\lbrace p^2, ~\max\left\lbrace\frac{|S||E|}{pk}, ~\frac{|S|^{\frac{1}{2}}|E|}{p^{\frac{1-ε}{2}}k^{\frac{1}{2}}}\right\rbrace \right\rbrace,$$ where $k$ bounds the radial multiplicity of $E$. For small sets $|E| \leq p$, we establish optimal bounds using weighted incidence theory. Analogous results are proved for the first Heisenberg group $\mathbb{H}_1(\mathbb{F}_p)$ acting on $\mathbb{F}_p^3$. Our approach reformulates the problem as an incidence question in a bipartite action graph. The proofs combine Fourier analytic techniques, energy estimates, point-line incidence bounds, and area-energy inequalities for skew dot products. The methods extend classical sum-product type problems and incidence theory to noncommutative group actions.

math.CO↗

arXiv2026

Synthesis and guided assembly of niobium trisulfide nanowires and nanowire chains by chemical vapor deposition

One-dimensional (1D) nanostructures of transition metal trichalcogenides (TMT) show unique properties through the combination of their anisotropic bonding and low dimensionality. Scalable synthesis approaches that enable control over the morphology, dimensions, and interfaces of 1D TMTs with other nanoscale materials could allow these properties to be used in novel devices. Here, we report chemical vapor deposition of a 1D TMT, namely niobium trisulfide (NbS3) in the form of nanowires, on different substrates, including bulk substrates (amorphous SiO2/Si and crystalline c-sapphire) and several two-dimensional (2D) van der Waals materials (graphene, h-BN, CrSBr). We demonstrate high growth yield with axial growth rates of up to 40 micrometer/min and with two different growth modes: short nanowires of rectangular cross-section, and unusual long, "chained nanowires" up to 100 micrometer in length with sawtooth morphology. We discuss a mechanism that accounts for the two morphologies and discuss how the structure can be tuned through substrate choice and growth conditions. We further demonstrate guided assembly at the edges of graphene and h-BN, as well as epitaxial growth on few-layer CrSBr and c-sapphire. These results open pathways to explore scalable synthesis and directed assembly of 1D TMT nanomaterials in unique morphologies.

cond-mat.mtrl-sci↗

arXiv2026

Orthogonal projections and incidence bounds in planes over prime order fields

Let $p$ be an odd prime and let $E\subset \mathbb{F}_p^2$ with $|E|=p^a$, where $0<a\le 1$. For a direction $V$ (a $1$-dimensional subspace of $\mathbb{F}_p^2$), let $π^V:\mathbb{F}_p^2\to \mathbb{F}_p^2/V$ denote the quotient map. We bound the size of the exceptional set of directions for which the projection $π^V(E)$ is small. More precisely, for $a/2\le s\le a$, define $$ T_s^{1,2}(E):=\{V\in G(1,\mathbb{F}_p^2):\ |π^V(E)|<p^s\}. $$ We prove $$ |T_s^{1,2}(E)|\ll \min\{\,p^{\frac52 s-a},\ p^{6s-3a},\ p^s\,\}, $$ which improves the best previously known estimates over prime fields in the range $a/2\le s<2a/3$, and yields the first substantial progress toward Chen's 2018 conjecture. The key new ingredient is a novel point-line incidence bound, of independent interest, that yields a power saving when the line set spans only moderately many distinct directions. In the reverse direction, we also obtain an incidence estimate for Cartesian products $A\times B$ with line families $\{y=ax+b:\ a,b\in C\}$ with explicit dependence on the additive energy $E^+(C)$. We also discuss connections to the sum-set problem and the distinct dot-product values conjecture.

math.CO↗