关于无平方因子 $N$ 的 $\mathbb{Z}/N\mathbb{Z}$ 中抛物线的 Hickman 与 Wright 限制问题
经典分析与常微分方程
2025-09-15 v1 组合数学
数论
摘要
Hickman 和 Wright 证明了 Z / N Z \mathbb{Z}/N\mathbb{Z} Z / N Z Σ \Sigma Σ L 2 L^2 L 2 ( 1 ∣ Σ ∣ ∑ m ∈ Σ ∣ f ^ ( m ) ∣ 2 ) 1 2 ≤ C ϵ N ϵ ⋅ N − 1 ( ∑ x ∈ ( Z / N Z ) 2 ∣ f ( x ) ∣ 6 5 ) 5 6 \left(\frac{1}{|\Sigma|}\sum\limits_{m\in\Sigma}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_\epsilon N^\epsilon\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{6}{5}\right)^\frac{5}{6} ( ∣Σ∣ 1 m ∈ Σ ∑ ∣ f ( m ) ∣ 2 ) 2 1 ≤ C ϵ N ϵ ⋅ N − 1 x ∈ ( Z / N Z ) 2 ∑ ∣ f ( x ) ∣ 5 6 6 5 f : ( Z / N Z ) 2 → C f:(\mathbb{Z}/N\mathbb{Z})^2\rightarrow \mathbb{C} f : ( Z / N Z ) 2 → C ϵ > 0 \epsilon>0 ϵ > 0 N N N N = p 2 N = p^2 N = p 2 p p p N = p N = p N = p ( 1 ∣ Σ ∣ ∑ m ∈ Σ ∣ f ^ ( m ) ∣ 2 ) 1 2 ≤ C N − 1 ( ∑ x ∈ ( Z / N Z ) 2 ∣ f ( x ) ∣ 4 3 ) 3 4 . \left(\frac{1}{|\Sigma|}\sum\limits_{m\in\Sigma}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4}. ( ∣Σ∣ 1 m ∈ Σ ∑ ∣ f ( m ) ∣ 2 ) 2 1 ≤ C N − 1 x ∈ ( Z / N Z ) 2 ∑ ∣ f ( x ) ∣ 3 4 4 3 . N N N ( 1 ∣ Σ ∣ ∑ m ∈ Σ ∣ f ^ ( m ) ∣ 2 ) 1 2 ≤ C ϵ N ϵ ⋅ N − 1 ( ∑ x ∈ ( Z / N Z ) 2 ∣ f ( x ) ∣ 4 3 ) 3 4 , \left(\frac{1}{|\Sigma|}\sum\limits_{m\in\Sigma}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_\epsilon N^\epsilon\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4}, ( ∣Σ∣ 1 m ∈ Σ ∑ ∣ f ( m ) ∣ 2 ) 2 1 ≤ C ϵ N ϵ ⋅ N − 1 x ∈ ( Z / N Z ) 2 ∑ ∣ f ( x ) ∣ 3 4 4 3 ,
引用
@article{arxiv.2509.09885,
title = {On a Restriction Problem of Hickman and Wright for the Parabola in $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$},
author = {Nathaniel Kingsbury-Neuschotz},
journal= {arXiv preprint arXiv:2509.09885},
year = {2025}
}
备注
10 pages