Improved discrete restriction for the parabola

改进的抛物线高低法

An improved high-low method for the parabola

本文改进Guth-Maldague-Wang高低法，通过正交展开和新型平方函数，获得抛物线离散限制常数尖锐估计及有理点加性能量无直径界，分别验证了近期两项研究成果。

关于无平方因子 $N$ 下 $\mathbb{Z}/N\mathbb{Z}$ 上抛物线的 Hickman 和 Wright 限制问题

On a Restriction Problem of Hickman and Wright for the Parabola over $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$

将 Mockenhaupt-Tao 关于素数 $N$ 的抛物线 $L^2$ 限制估计推广到无平方因子 $N$，证明了一个更强的界，并讨论了在不确定性原理和信号恢复中的应用。

Discrete restriction estimates of epsilon-removal type for kth-powers and k-paraboloids

We obtain restriction estimates of $ε$-removal type for the set of $k$-th powers of integers, and for discrete $d$-dimensional surfaces of the form \[ \{ (n_1,\dots,n_d,n_1^k + \dotsb + n_d^k) \,:\, |n_1|,\dots,|n_d| \leq N \}, \] which we term '$k$-paraboloids'. For these surfaces, we obtain a satisfying range of exponents for large values of $d,k$. We also obtain estimates of $ε$-removal type in the full supercritical range for $k$-th powers and for $k$-paraboloids of dimension $d < k(k-2)$. We rely on a variety of techniques in discrete harmonic analysis originating in Bourgain's works on t…

有限域上的一个改进的 $L^2$ 限制定理

An improved $L^2$ restriction theorem in finite fields

本文将有限域 Stein–Tomas 型 $L^2$ 限制定理中 Fourier 变换的一致界改为 $L^p$ 界，从而在多种情形下扩展 Mockenhaupt–Tao 指数范围，并给出 Sidon 集与 Hamming 簇应用。

Diophantine approximation with prime restriction in function fields

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $pα$ when $α$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to the nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered t…

Maass形式在超曲面和测地管上的限制

Restrictions of Maass forms on $\mathrm{SL}(2,\mathbb{C})$ to hyperbolic surfaces and geodesic tubes

本文通过算术放大方法改进了Maass形式在超曲面和测地管上的L²界，获得了更优的功率节省结果。