• 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.

1. 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)

2. 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)

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

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

1. Lp -integrability of functions with Fourier support on a fractal set of moment curve (2024), in preparation.
   (with S. Duan and D. Ryou)

2. 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)

3. Discretized sum-product type problems: Energy variants and Applications, (2022), submitted, [arXiv].
   (with A. Mohammadi, T. Pham, and C-Y. Shen)