## Research

### Research interest:

- Harmonic Analysis, Geometric Measure Theory, Discrete Geometry, and Combinatorial Number Theory

One of the most important questions in Geometric measure theory and Combinatorics is: How large does a given set in vector space (F_q^d or R^d) need to be to make sure that it contains copies of a given configuration of points?

For example, I am interested in Falconer distance conjecture and related problems.

Falconer distance conjecture says that for any compact set A ⊂ R^d of Hausdorff dimension greater than d/2, the distance set ∆(A) is of positive Lebesgue measure. The best-known result in plane due to Guth, Iosevich, Ou, and Wang (2019) gave the exponent 5/4, and the conjecture is still open.

### My Co-Authors (11)

### Publications:

Pinned simplices and connections to product of sets on paraboloids, Indiana Univ. Math. J. (2024), to appear, [arXiv].

(with A. Iosevich, T. Pham, and C-Y. Shen)Structural theorems on the distance sets over finite fields, Forum Math. 35 (2023), no.4, 925-938, [arXiv], [journal].

(with D. Koh and T. Pham)On the k-resultant modulus set problem on varieties over finite fields, Int. J. Number Theory 19 (2023), no.3, 569-579, [arXiv], [journal].

Bound for volumes of sub-level sets of polynomials and applications to singular integrals, (2020), [arXiv].

(with Loi Ta Le)

### Preprints:

Lp -integrability of functions with Fourier support on a fractal set of moment curve (2024), in preparation.

(with S. Duan, A. Iosevich, and D. Ryou)Packing sets in Euclidean space by affine transformations (2024), in preparation.

(with A. Iosevich, P. Mattila, E. Palsson, T. Pham, S. Senger, and C-Y. Shen)Discretized sum-product type problems: Energy variants and Applications, (2022), submitted, [arXiv].

(with A. Mohammadi, T. Pham, and C-Y. Shen)