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

One of the most important questions in Geometric measure theory is: How large does a given set in vector space ( 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. Lp -integrability of functions with Fourier supports on fractal sets on moment curve, with S. Duan and D. Ryou (2024), submitted.

2. Packing sets in Euclidean space by affine transformations, with A. Iosevich, P. Mattila, E. Palsson, T. Pham, S. Senger, and C-Y. Shen (2024), submitted.

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

1. Pinned simplices and connections to product of sets on paraboloids, with A. Iosevich, T. Pham, and C-Y. Shen, Indiana Univ. Math. J. (2024), to appear.

2. Structural theorems on the distance sets over finite fields, with D. Koh and T. Pham, Forum Math. 35 (2023), no.4, 925-938.

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

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