Research interest

Selected publications

(Geometric Measure Theory/Additive Combinatorics)

We introduce Frostman conditions for bivariate random variables and study discretized entropy sum-product phenomena in both independent and dependent regimes. We prove that for any non-degenerate rational quadratic form $\phi(x, y)$ and parameter $0 < s < 1$, there exists a positive constant $\epsilon = \epsilon(\phi,s)$ such that

$$\max\{H_n(X+Y), H_n(\phi(X,Y))\} \geq n(s+\epsilon)$$

for sufficiently large $n$, where the precise conditions on $(X,Y)$ depend on the Frostman level. The proof introduces a novel multi‑step entropy framework, combining the submodularity formula, the discretized entropy Balog-Szemer\'{e}di-Gowers theorem, and state-of-the-art results on the Falconer distance problem, to reduce general forms to a diagonal core case. 

As an application, we derive a result on a discretized sum-product type problem. In particular, for a $\delta$-separated set $A\subset [0, 1]$ of cardinality $\delta^{-s}$, satisfying some non-concentration conditions, and a dense subset $G\subset A\times A$, there exists $\epsilon=\epsilon(s, \phi)>0$ such that    

$$E_\delta(A+_GA) + E_\delta(\phi_G(A, A)) \gg\delta^{-\epsilon}(\#A)$$

for all $\delta$ small enough. Here by $\mathcal{E}_\delta(A)$ we mean the $\delta$-covering number of $A$, $\{A+_GA:=\{x+y\colon (x, y)\in G\}$, and $\phi_G(A):=\{\phi(x, y)\colon (x, y)\in G\}$.

(Combinatorial Number Theory)

In this paper, we studied the distance problem in the setting of finite $p$-adic rings. In odd dimensions, our results are essentially sharp. In even dimensions, we clarified the conjecture and provided supporting examples. Surprisingly, unlike in the finite field case, we identified a large family of sets in this setting for which the distance conjecture holds. By developing new restriction type estimates associated with circles and orbits, and with a group theoretic argument, we proved the $4/3$-parallel result in two dimensions, addressing a question posed by Alex Iosevich. Our main results improve and extend the recent work of Ben Lichtin (2019, 2023).

(Geometric Measure Theory)

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 proved that the image set $f(A, B, C)$ is of positive Lebesgue measure. This result is optimal, and the proof highlights connections between Combinatorics and Geometric Measure Theory. As an application, we obtained an improvement on the Falconer distance problem for Cartesian product sets in three dimensions.

(Discrete Geometry/Combinatorial Number Theory)

Let $A$ and $B$ be sets in a finite vector space. In this paper, we studied the magnitude of the set $A\cap f(B)$, where $f$ runs through a set of transformations. Specifically, we focused on the cases that the set of transformations is given by rigid-motions or orthogonal projections. One of the most important contributions of this paper is to show that if $A, B\subset \mathbb{F}_q^d$ satisfy some natural conditions then, for almost every $g\in O(d)$, there are at least $\gg q^d$ elements $z\in \mathbb{F}_q^d$ such that 

\[|A\cap (g(B)+z)| \sim \frac{|A||B|}{q^d}.\]

This implies that $|A-gB|\gg q^d$ for almost every $g\in O(d)$. In the flavor of expanding functions, with $|A|\le |B|$, we also show that the image $A-gB$ grows exponentially. In two dimensions, the result simply says that if $|A|=q^x$ and $|B|=q^y$, as long as $0<x\le y<2$, then for almost every $g\in O(2)$, we can always find $\epsilon=\epsilon(x, y)>0$ such that $|A-gB|\gg |B|^{1+\epsilon}$. To prove these results, we need to develop new and robust incidence bounds between points and rigid motions by using a number of techniques including algebraic methods and discrete Fourier analysis. Our results are essentially sharp in odd dimensions. In the prime field plane, we further employ recent L^2 distance bounds and point--line/plane incidence machinery to derive improvements. Notable applications include a strong prime field analogue of a question of Mattila related to the Falconer distance problem, the Rotational Erdős-Falconer distance problem, and a quadratic expansion law. Taken together, the results in this paper present a robust two-way link between intersection phenomena and distance problems over finite fields, with dimension-uniform consequences and sharpness in several ranges.

(Restriction theory/Geometric Measure Theory)

In this paper, we solved completely the finite field cone restriction conjecture in four dimensions  with $-1$ non-square, and introduced a new approach to study incidence problems via restriction theory.

(Discrete Geometry)

In this paper, we obtained the exponent $5/4$ for the Erd\H{o}s-Falconer distance problem in $\mathbb{F}_p^2$, and the exponent $2/3$ for the pinned distinct distances over arbitrary fields. These are the current best exponents in the literature. 

(Discrete Geometry/Combinatorial Number Theory)

In this paper, we proved the Elekes--R\'{o}nyai's conjecture for quadratic polynomials in three variables over arbitrary fields. More precisely, we showed that any quadratic polynomial in three variables that is not of the form $g(h(x)+k(y)+l(z))$ is an expander. 

(Discrete Geometry/Combinatorial Number Theory)

In this paper, we proved two generalizations of the Schwartz-Zippel lemma for non-Cartesian varieties of dimensions smaller than or equal to three.  The bound for three-dimensional varieties generalizes the famous Szemerédi-Trotter theorem.

All publications

Ph. D. Thesis: Erdos Distinct Distances Problem and Extensions over Finite Spaces, EPFL, Lausanne, 2017. 

My Co-Authors (52):

Dao Nguyen Van Anh, Tanbir Ahmed, Arie Bialostocki, Pablo Bhowmick, Philipp Birklbauer, Paige Bright,  Daewoong Cheong, Francois Clement, Ye ̧sim Demiro ̆glu Karabulut, Le Quang Hung, Norbert Hegyvari, Le Quang Ham, Do Duy Hieu, Pham Duc Hiep, Alex Iosevich, Doowon Koh,  Jeong Han Kim, Pertti Mattila,  Brendan Murphy, Mojgan Mirzaei, Ben Lund, Sujin Lee,  Ali Mohammadi, Hossein N. Mojarrad, Janos Pach, Eyvi Palsson, Quy Pham, Giorgis Petridis,  Jonathan Pakianathan, Nguyen Duy Phuong, Misha Rudnev, Steven Senger, Nguyen Minh Sang, Andrew Suk, Kaloyan Slavov,  Sophie Stevens, Chun-Yen Shen, Dung The Tran, Michael Tait, Craig Timmons, Gabor Tardos, Nguyen Trung-Tuan, Nguyen Duc-Thang,  Vu Thi Huong Thu, Nguyen Pham Minh Tri,  Le Anh Vinh, Claudiu Valculescu, Robert Won, Audie Warren, Boqing Xue, Semin Yoo, Frank de Zeeuw.

Referee Service: 

Annales Fennici Mathematici, Computational Geometry, Discrete and Computational Geometry (DCG),  Discrete Mathematics, European Journal of Combinatorics (EJC), SIAM Journal on Discrete Mathematics (SIDMA), Journal of Combinatorial Theory Series A (JCTA), Discrete Applied Mathematics (DAM), Symposium on Computational Geometry (SoCG), Graphs and Combinatorics,  Forum Mathematicum (FORUM), Journal of Fourier Analysis and Applications (JFAA), New York  Journal of Mathematics, Finite Fields and Their Applications (FFA), Journal of Number Theory, Proceedings of the American Mathematical Society (PAMS), Vietnam Journal of Mathematics, Math Reviews, ZbMATH