arXiv subjects
Publications and source records attributed to Thang Pham.
In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in $SL_2(\mathbb{F}_p)$ with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyvári and Hennecart.
We report the synthesis of the quasi-one-dimensional transition metal trichalcogenide NbSe3 in the few chain limit, including the realization of isolated single chains. The chains are encapsulated in protective boron nitride or carbon nanotube sheaths to prevent oxidation and to facilitate characterization. Transmission electron microscopy reveals static and dynamic structural torsional waves not found in bulk NbSe3 crystals. Electronic structure calculations indicate that charge transfer drives the torsional wave instability. Very little covalent bonding is found between the chains and the nanotube sheath, leading to relatively unhindered longitudinal and torsional dynamics for the encapsulated chains.
Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough.
In this short note, we study the distribution of spreads in a point set $\mathcal{P} \subseteq \mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $\varepsilon > 0$, if $|\mathcal{P}| \geq (1+\varepsilon) q^{\lceil d/2 \rceil}$, then $\mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $\mathcal{P} \subset \mathbb{F}_q^d$ of size $|\mathcal{P}| = q^{\lceil d/2 \rceil}$ that determine at most one spread.
We prove bounds on intersections of algebraic varieties in $\mathbb{C}^4$ with Cartesian products of finite sets from $\mathbb{C}^2$, and we point out connections with several classic theorems from combinatorial geometry. Consider an algebraic variety $X$ in $\mathbb{C}^4$ of degree $d$, such that the polynomials defining $X$ are not all of the form $F(x,y,s,t) = G(x,y)H(x,y,s,t) + K(s,t)L(x,y,s,t)$. Let $P$ and $Q$ be finite subsets of $\mathbb{C}^2$ of size $n$. If $X$ has dimension one or two, then we prove $|X\cap (P\times Q)| = O_d(n)$, while if $X$ has dimension three, then $|X\cap (P\times Q)| =O_{d,\varepsilon}(n^{4/3+\varepsilon})$ for any $\varepsilon>0$. Both bounds are best possible in this generality (except for the $\varepsilon$). These bounds can be viewed as different generalizations of the Schwartz-Zippel lemma, where we replace a product of "one-dimensional" finite subsets of $\mathbb{C}$ by a product of "two-dimensional" finite subsets of $\mathbb{C}^2$. The bound for three-dimensional varieties generalizes the Szemerédi-Trotter theorem. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries of our two bounds, we obtain bounds on the number of repeated and distinct values of polynomials and polynomial maps of pairs of points in $\mathbb{C}^2$, with a characterization of those maps for which no good bounds hold. These results generalize known bounds on repeated and distinct Euclidean distances.
Hegyvári and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $B\cdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.
In this short note, we use Rudnev's point-plane incidence bound to improve some results on conditional expanding bounds for two-variable functions over arbitrary fields due to Hegyvári and Hennecart.
We determine which quadratic polynomials in three variables are expanders over an arbitrary field $\mathbb{F}$. More precisely, we prove that for a quadratic polynomial $f\in \mathbb{F}[x,y,z]$, which is not of the form $g(h(x)+k(y)+l(z))$, we have $|f(A\times B\times C)|\gg N^{3/2}$ for any sets $A,B,C\subset \mathbb{F}$ with $|A|=|B|=|C|=N$, with $N$ not too large compared to the characteristic of $\mathbb{F}$. We give several applications. We use this result for $f=(x-y)^2+z$ to obtain new lower bounds on $|A+A^2|$ and $\max\{|A+A|,|A^2+A^2|\}$, and to prove that a Cartesian product $A\times\cdots \times A\subset \mathbb{F}^d$ determines almost $|A|^2$ distinct distances if $|A|$ is not too large.
Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is large odd prime power. In this paper, we improve some recent results on the additive energy of the distance set, and on sumsets of the distance set due to Shparlinski (2016). More precisely, we prove that for $\mathcal{E}\subseteq \mathbb{F}_q^d$, if $d=2$ and $q^{1+\frac{1}{4k-1}}=o(|\mathcal{E}|)$ then we have $|kΔ_{\mathbb{F}_q}(\mathcal{E})|=(1-o(1))q$; if $d\ge 3$ and $q^{\frac{d}{2}+\frac{1}{2k}}=o(|\mathcal{E}|)$ then we have $|kΔ_{\mathbb{F}_q}(\mathcal{E})|=(1-o(1))q,$ where $kΔ_{\mathbb{F}_q}(\mathcal{E}):=Δ_{\mathbb{F}_q}(\mathcal{E})+\cdots+Δ_{\mathbb{F}_q}(\mathcal{E}) ~(\mbox{$k$ times}).$
We demonstrate the fabrication of individual nanopores in hexagonal boron nitride (hBN) with atomically precise control of the pore size. Previous methods of pore production in other 2D materials create pores of irregular geometry with imprecise diameters. By taking advantage of the preferential growth of boron vacancies in hBN under electron beam irradiation, we are able to observe the pore growth via transmission electron microscopy, and terminate the process when the pore has reached its desired size. Careful control of beam conditions allows us to nucleate and grow individual triangular and hexagonal pores with diameters ranging from subnanometer to 6nm over a large area of suspended hBN using a conventional TEM. These nanopores could find application in molecular sensing, DNA sequencing, water desalination, and molecular separation. Furthermore, the chemical edge-groups along the hBN pores can be made entirely nitrogen terminated or faceted with boron-terminated edges, opening avenues for tailored functionalization and extending the applications of these hBN nanopores.
In this paper, we use methods from spectral graph theory to obtain some results on the sum-product problem over finite valuation rings $\mathcal{R}$ of order $q^r$ which generalize recent results given by Hegyvári and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$, if $A$ and $B$ are two sets in $\mathcal{R}^*$ with $|A|=|B|=q^α$, then \[\max\left\lbrace |f(A, B)|, |g(A, B)| \right\rbrace\gtrsim |A|^{1+Δ(α)},\] for some $Δ(α)>0$.
We prove a Szemerédi-Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and Sárközy on the solvability of the equations $a + b = cd$ and $ab + 1 = cd$ over a finite field. Other analogous results that are known to hold in finite fields are generalized to finite quasifields.
Let $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if $|\mathcal{E}|\gg q^{d-\frac{d-1}{k+1}}$ then $\mathcal{E}$ determines a positive proportion of all $k$-simplices. In this paper, we give an improvement of this result in the case when $\mathcal{E}$ is the Cartesian product of sets. More precisely, we show that if $\mathcal{E}$ is the Cartesian product of sets and $q^{\frac{kd}{k+1-1/d}}=o(|\mathcal{E}|)$, the number of congruence classes of $k$-simplices determined by $\mathcal{E}$ is at least $(1-o(1))q^{\binom{k+1}{2}}$, and in some cases our result is sharp.
Organic-inorganic halide perovskite materials have emerged as attractive alternatives to conventional solar cell building blocks. Their high light absorption coefficients and long diffusion lengths suggest high power conversion efficiencies (PCE),1-5 and indeed perovskite-based single band gap and tandem solar cell designs have yielded impressive performances.1-16 One approach to further enhance solar spectrum utilization is the graded band gap, but this has not been previously achieved for perovskites. In this study, we demonstrate graded band gap perovskite solar cells with steady-state conversion efficiencies averaging 18.4%, with a best of 21.7%, all without reflective coatings. An analysis of the experimental data yields high fill factors of ~75% and high short circuit current densities up to 42.1 mA/cm2. These cells, which are based on a novel architecture of two perovskite layers (MASnI3 and MAPbI3-xBrx), incorporating GaN, monolayer hexagonal boron nitride, and graphene aerogel, display the highest efficiency ever reported for perovskite solar cells.
In this paper we introduce a unified approach to deal with incidence problems between points and varieties over finite fields. More precisely, we prove that the number of incidences $I(\mathcal{P}, \mathcal{V})$ between a set $\mathcal{P}$ of points and a set $\mathcal{V}$ of varieties of a certain form satisfies $$\left\vert I(\mathcal{P},\mathcal{V})-\frac{|\mathcal{P}||\mathcal{V}|}{q^k}\right\vert\le q^{dk/2}\sqrt{|\mathcal{P}||\mathcal{V}|}.$$ This result is a generalization of the results of Vinh (2011), Bennett et al. (2014), and Cilleruelo et al. (2015). As applications of our incidence bounds, we obtain results on the pinned value problem and the Beck type theorem for points and spheres. Using the approach introduced, we also obtain a result on the number of distinct distances between points and lines in $\mathbb{F}_q^2$, which is the finite field analogous of a recent result of Sharir et al. (2015).
We use methods from spectral graph theory to obtain bounds on the number of incidences between $k$-planes and $h$-planes in $\mathbb{F}_q^d$ which generalize a recent result given by Bennett, Iosevich, and Pakianathan (2014). More precisely, we prove that the number of incidences between a set $\mathcal{P}$ of $k$-planes and a set $\mathcal{H}$ of $h$-planes with $h\ge 2k+1$, which is denoted by $I(\mathcal{P},\mathcal{H})$, satisfies \[\left\vert I(\mathcal{P},\mathcal{H})-\frac{|\mathcal{P}||\mathcal{H}|}{q^{(d-h)(k+1)}}\right\vert \lesssim q^{\frac{(d-h)h+k(2h-d-k+1)}{2}}\sqrt{|\mathcal{P}||\mathcal{H}|}. \]