Xuancheng Shao

activity

20242026

collaborators

5 papers

math.CO2026

Higher additive energies on discrete cubes

Xuancheng Shao, Yu-Chen Sun

Let $m, n \geq 2$ be integers. We study the least exponent $t_{m, n}$ such that the $m$ -fold additive energy of any subset $A$ of the discrete cube ${0, 1, \dots, n - 1}^{d}$ in any dimens…

math.NT2026

Linear equations in Piatetski-Shapiro primes

Xuancheng Shao, Yu-Chen Sun

We establish discorrelation estimates between the Piatetski-Shapiro prime set \[ \mathcal{P}_γ := \{p \text{ is prime and } p = \lfloor n^{1/γ} \rfloor \text{ for some } n \in \mat…

math.NT2026

The sum-product phenomenon for dense subsets of finite fields

Xuancheng Shao

Let $F_{p}$ be a finite field of prime order $p$ and let $A \subset F_{p}$ be a subset. In the dense regime when $∣ A ∣ \geq α p$ for some $α \in (0, 1)$ , we determine the…

math.NT2025

Quantitative bounds in a popular polynomial Szemerédi theorem

Xuancheng Shao, Mengdi Wang

We obtain polylogarithmic bounds in the polynomial Szemerédi theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mat…

math.NT2024

Sums of Kloosterman sums over square-free and smooth integers

Xuancheng Shao, Igor E. Shparlinski, Laurence P. Wijaya

Recently there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consi…