AI 中文总结
该研究针对有限域上的多项式与乘法子群，建立了大子集存在满足相邻元素多项式值属于k次幂集合的循环排列的阈值，并给出该阈值的上下界。
AI 中文摘要
设f(t)∈ℤ[t]为非零判别式的非恒定多项式，k≥2为整数。对每个足够大的素数p≡1 mod k，通过应用有限域上的混合指数和、离散傅里叶分析与谱图理论，我们建立阈值c(p,k,f)，使得满足#A≥c(p,k,f)的任意子集A⊆𝔽_p，都存在A的排列a₁,a₂,…,a_{#A}，对任意1≤i≤#A满足f(a_i+a_{i+1})∈{xᵏ:x∈𝔽_p^*}（其中a_{#A+1}=a₁）。我们还给出该最小阈值的上下界。
英文摘要
Let $f(t)\in\mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv1\pmod{k}$, we study cyclic orderings of subsets $A\subseteq \mathbb{F}_p$ for which $ f(a_i+a_{i+1})$ is a nonzero $k$-th power for every consecutive pair. By combining mixed character-sum estimates, Fourier analysis on $\mathbb{F}_p$, and spectral graph methods, we establish a threshold $c(p,k,f)$ such that every subset $A$ with $\#A\ge c(p,k,f)$ admits such a cyclic ordering. We also give lower and upper bounds for the optimal threshold.
Comments21 pages. Comments are welcome
