## 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 and D. Ryou)Packing sets in Euclidean space by affine transformations (2024), submitted, [arXiv].

(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)