arXiv subjects

Thang Pham

Publications and source records attributed to Thang Pham.

At least 73 records · Page 4Linked to original sources

arXiv2022

Radial projection theorems in finite spaces

Motivated by recent results on radial projections and applications to the celebrated Falconer distance problem, we study radial projections in the setting of finite fields. More precisely, we extend results due to Mattila and Orponen (2016), Orponen (2018), and Liu (2020) to finite spaces. In some cases, our results are stronger than the corresponding results in the continuous setting. In particular, we solve the finite field analog of a conjecture due to Liu and Orponen on the exceptional set of radial projections of a set of dimension between $d-2$ and $d-1$.

math.CA↗

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↗

arXiv2022

A Point-Conic Incidence Bound and Applications over $\mathbb F_p$

In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These include new lower bounds on the number of pinned algebraic distances as well as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on the size of the distance set formed by two large subsets of finite dimensional vector spaces over finite fields. We also prove a variant of Beck's theorem for conics.

math.CO↗

arXiv2022

Suspended graphene membranes to control Au nucleation and growth

Control of nucleation sites is an important goal in materials growth: nuclei in regular arrays may show emergent photonic or electronic behavior, and once the nuclei coalesce into thin films, the nucleation density influences parameters such as surface roughness, stress, and grain boundary structure. Tailoring substrate properties to control nucleation is therefore a powerful tool for designing functional thin films and nanomaterials. Here, we examine nucleation control for metals deposited on two-dimensional (2D) materials in a situation where substrate effects are absent and heterogeneous nucleation sites are minimized. Through quantification of faceted, epitaxial Au island nucleation on graphene, we show that ultra-low nucleation densities with nuclei several micrometers apart can be achieved on suspended graphene under conditions where we measure the nucleation density to be 2-3 orders of magnitude higher on the adjacent supported substrate. We estimate diffusion distances using nucleation theory and find a strong sensitivity of nucleation and diffusion to suspended graphene thickness. We suggest that nucleation site density control via substrate tuning may act as a platform for applications where non-lithographic patterning of metals on graphene is beneficial.

cond-mat.mtrl-sci↗

arXiv2022

Direct Imaging and Electronic Structure Modulation of Moiré Superlattices at the 2D/3D Interface

The atomic structure at the interface between two-dimensional (2D) and three-dimensional (3D) materials influences properties such as contact resistance, photo-response, and high-frequency electrical performance. Moiré engineering is yet to be utilized for tailoring this 2D/3D interface, despite its success in enabling correlated physics at 2D/2D interfaces. Using epitaxially aligned MoS2/Au{111} as a model system, we demonstrate the use of advanced scanning transmission electron microscopy (STEM) combined with a geometric convolution technique in imaging the crystallographic 32 A moiré pattern at the 2D/3D interface. This moiré period is often hidden in conventional electron microscopy, where the Au structure is seen in projection. We show, via ab initio electronic structure calculations, that charge density is modulated according to the moiré period, illustrating the potential for (opto-)electronic moiré engineering at the 2D/3D interface. Our work presents a general pathway to directly image periodic modulation at interfaces using this combination of emerging microscopy techniques.

cond-mat.mtrl-sci↗

arXiv2022

Structural theorems on the distance sets over finite fields

Let $\mathbb{F}_q$ be a finite field of order $q$. Iosevich and Rudnev (2005) proved that for any set $A\subset \mathbb{F}_q^d$, if $|A|\gg q^{\frac{d+1}{2}}$, then the distance set $Δ(A)$ contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be $\frac{d}{2}$ in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture is still wide open in higher dimensions. To fill the gap, we need to understand more about the structures of the distance sets, the main purpose of this paper is to provide some structural theorems on the distribution of square and non-square distances.

math.NT↗

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↗

arXiv2021

Configurations of rectangles in a set in $\mathbb{F}_q^2$

Let $\mathbb{F}_q$ be a finite field of order $q$. In this paper, we study the distribution of rectangles in a given set in $\mathbb{F}_q^2$. More precisely, for any $0<δ\le 1$, we prove that there exists an integer $q_0=q_0(δ)$ with the following property: if $q\ge q_0$ and $A$ is a multiplicative subgroup of $\mathbb{F}^*_q$ with $|A|\ge q^{2/3}$, then any set $S\subset \mathbb{F}_q^2$ with $|S|\ge δq^2$ contains at least $\gg \frac{|S|^4|A|^2}{q^5}$ rectangles with side-lengths in $A$. We also consider the case of rectangles with one fixed side-length and the other in a multiplicative subgroup $A$.

math.CO↗

arXiv2021

A point-sphere incidence bound in odd dimensions and applications

In this paper, we prove a new point-sphere incidence bound in vector spaces over finite fields. More precisely, let $P$ be a set of points and $S$ be a set of spheres in $\mathbb{F}_q^d$. Suppose that $|P|, |S|\le N$, we prove that the number of incidences between $P$ and $S$ satisfies \[I(P, S)\le N^2q^{-1}+q^{\frac{d-1}{2}}N,\] under some conditions on $d, q$, and radii. This improves the known upper bound $N^2q^{-1}+q^{\frac{d}{2}}N$ in the literature. As an application, we show that for $A\subset \mathbb{F}_q$ with $q^{1/2}\ll |A|\ll q^{\frac{d^2+1}{2d^2}}$, one has \[\max \left\lbrace |A+A|,~ |dA^2|\right\rbrace \gg \frac{|A|^d}{q^{\frac{d-1}{2}}}.\] This improves earlier results on this sum-product type problem over arbitrary finite fields.

math.CO↗

arXiv2021

Distribution of distances in five dimensions and related problems

In this paper, we study the Erdős-Falconer distance problem in five dimensions for sets of Cartesian product structures. More precisely, we show that for $A\subset \mathbb{F}_p$ with $|A|\gg p^{\frac{13}{22}}$, then $Δ(A^5)=\mathbb{F}_p$. When $|A-A|\sim |A|$, we obtain stronger statements as follows: If $|A|\gg p^{\frac{13}{22}}$, then $(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$ If $|A|\gg p^{\frac{4}{7}}$, then $(A-A)^2+(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$ We also prove that if $p^{4/7}\ll |A-A|=K|A|\le p^{5/8}$, then \[|A^2+A^2|\gg \min \left\lbrace \frac{p}{K^4}, \frac{|A|^{8/3}}{K^{7/3}p^{2/3}}\right\rbrace.\] As a consequence, $|A^2+A^2|\gg p$ when $|A|\gg p^{5/8}$ and $K\sim 1$, where $A^2=\{x^2\colon x\in A\}$.

math.CO↗

arXiv2021

Atomistic spin textures on-demand in the van der Waals layered magnet CrSBr

Controlling magnetism in low dimensional materials is essential for designing devices that have feature sizes comparable to several critical length scales that exploit functional spin textures, allowing the realization of low-power spintronic and magneto-electric hardware. [1] Unlike conventional covalently-bonded bulk materials, van der Waals (vdW)-bonded layered magnets [2-4] offer exceptional degrees of freedom for engineering spin textures. [5] However, their structural instability has hindered microscopic studies and manipulations. Here, we demonstrate nanoscale structural control in the layered magnet CrSBr creating novel spin textures down to the atomic scale. We show that it is possible to drive a local structural phase transformation using an electron beam that locally exchanges the bondings in different directions, effectively creating regions that have vertical vdW layers embedded within the horizontally vdW bonded exfoliated flakes. We calculate that the newly formed 2D structure is ferromagnetically ordered in-plane with an energy gap in the visible spectrum, and weak antiferromagnetism between the planes. Our study lays the groundwork for designing and studying novel spin textures and related quantum magnetic phases down to single-atom sensitivity, potentially to create on-demand spin Hamiltonians probing fundamental concepts in physics, [6-10] and for realizing high-performance spintronic, magneto-electric and topological devices with nanometer feature sizes. [11,12]

cond-mat.mes-hall↗

arXiv2021

On the finite field cone restriction conjecture in four dimensions and applications in incidence geometry

The first purpose of this paper is to solve completely the finite field cone restriction conjecture in four dimensions with $-1$ non-square. The second is to introduce a new approach to study incidence problems via restriction theory. More precisely, using the cone restriction estimates, we will prove sharp point-sphere incidence bounds associated with complex-valued functions for sphere sets of small size. Our incidence bounds with a specific function improve significantly a result given by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev.

math.CA↗

arXiv2021

On the Pinned Distances Problem in Positive Characteristic

We study the Erd\H os-Falconer distance problem for a set $A\subset \mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$ determines an asymptotically full proportion of the feasible $p$ distances. For small sets $A$, namely when $|A|\leq p^{4/3}$ over any $\mathbb{F}$, we prove that either $A$ determines $\gg|A|^{2/3}$. For both large and small sets, the results proved are in fact for pinned distances.

math.CO↗

arXiv2021

An energy decomposition theorem for matrices and related questions

Given $A\subseteq GL_2(\mathbb{F}_q)$, we prove that there exist disjoint subsets $B, C\subseteq A$ such that $A = B \sqcup C$ and their additive and multiplicative energies satisfying \[ \max\{\,E_{+}(B),\, E_{\times}(C)\,\}\ll \frac{|A|^3}{M(|A|)}, \] where \begin{equation*} \label{eqn:MAminBVPolyLSSS} M(|A|) = \min\Bigg\{\,\frac{q^{4/3}}{|A|^{1/3}(\log|A|)^{2/3}},\, \frac{|A|^{4/5}}{q^{13/5}(\log|A|)^{27/10}}\,\Bigg\}. \end{equation*} We also study some related questions on moderate expanders over matrix rings, namely, for $A, B, C\subseteq GL_2(\mathbb{F}_q)$, we have \[|AB+C|, ~|(A+B)C|\gg q^4,\] whenever $|A||B||C|\gg q^{10 + 1/2}$. These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh (2019).

math.CO↗

arXiv2021

On the Mattila-Sjölin distance theorem for product sets

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 $d\ge 5$.

math.CA↗

arXiv2021

Falconer type functions in three variables

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.

math.CA↗

arXiv2021

An inverse-type problem for cycles in local Cayley distance graphs

Let $E$ be a proper symmetric subset of $S^{d-1}$, and $C_{\mathbb{F}_q^d}(E)$ be the Cayley graph with the vertex set $\mathbb{F}_q^d$, and two vertices $x$ and $y$ are connected by an edge if $x-y\in E$. Let $k\ge 2$ be a positive integer. We show that for any $α\in (0, 1)$, there exists $q(α, k)$ large enough such that if $E\subset S^{d-1}\subset \mathbb{F}_q^d$ with $|E|\ge αq^{d-1}$ and $q\ge q(α, k)$, then for each vertex $v$, there are at least $c(α, k)q^{\frac{(2k-1)d-4k}{2}}$ cycles of length $2k$ with distinct vertices in $C_{\mathbb{F}_q^d}(E)$ containing $v$. This result is the inverse version of a recent result due to Iosevich, Jardine, and McDonald (2021).

math.CO↗

arXiv2020

One-Dimensional Twisted and Tubular Structures of Zinc Oxide by Semiconductor-Catalyzed Vapor-Liquid-Solid Synthesis

The exploration of new catalysts for the vapor-liquid-solid (VLS) synthesis of one-dimensional (1-D) materials promises to yield new morphologies and functionality. Here, we show, for the model ZnO system, that this possible using a semiconductor (Ge) catalyst. In particular, two unusual morphologies are described: twisted nanowires and twisted nanotubes, in addition to the usual straight nanowires. The twisted nanotubes show large hollow cores and surprisingly high twisting rates (up to 9o/μm), which cannot be easily explained through the Eshelby twist model. A combination of ex situ and in situ transmission electron microscopy measurements suggest that the hollow core results from a competition between growth and etching at the Ge-ZnO interface during synthesis. The twisting rate is consistent with a softening of elastic rigidity. These results indicate that the use of unconventional, nonmetallic catalysts provide opportunities to synthesize unusual oxide structures with potentially useful properties.

cond-mat.mtrl-sci↗