Research and Publications

Certain problems in analytic number theory, e.g. the Gauss Circle Problem, can be reduced to studying the pointwise behavior of exponential sums on the integer ring. To understand an exponential sum, we apply some transformations and relate the consequent exponential sum to its L^p norm. This idea is originated from Vinogradov's method for bounding the Weyl sum. Afterwards we need good estimates of the L^p norm of certain exponential sums. 

The recent progress in harmonic analysis, known as decoupling theory, provides a powerful tool for estimating the norms of exponential sums. Two celebrated results are the pointwise estimate of the Riemann Zeta function on the critical line and Bourgain-Demeter-Guth's solution to Vinogradov's Mean Value Conjecture. My research is mainly about building connections between pointwise and average behaviors of exponential sums as well as deriving effective bounds on the norms of exponential sums.

I am also interested in two problems in additive combinatorics. One is bound for 3-term arithmetic progressions. The other one is Sárközy's problem, which seeks for upper bounds on sets with no square differences. The links contain updated progresses towards these two problems. 

Graduate Research:

5. An improvement on Gauss's Circle Problem and Dirichlet's Divisor Problem (with Xiaochun Li). Preprint

Undergraduate Research:

4. Extreme values of the derivative of Blaschke products and hypergeometric polynomials (with Leonid Kovalev), Bull. Sci. Math. 169 (2021), 102979. Journal Preprint

3. Near-isometric duality of Hardy norms with applications to harmonic mappings (with Leonid Kovalev), J. Math. Anal. Appl. 487 (2020), no. 2, 124040. Journal Preprint

2. Algebraic structure of the range of a trigonometric polynomial (with Leonid Kovalev), Bull. Aust. Math. Soc. 102 (2020), no. 2, 251-260. Journal Preprint

1. Fourier series of circle embeddings (with Leonid Kovalev), Comput. Methods Funct. Theory 19 (2019), no. 2, 323-340. Journal Preprint

Selected Talks:

1. An Improvement on Gauss’s Circle Problem and Dirichlet’s Divisor Problem, IASM of HIT, Dec. 20, 2023. Video Link