发表机构
Iowa State University（爱荷华州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限阿贝尔群上的会议信号，利用辛几何工具证明其存在性，并首次给出合数及素数阶循环群上的显式非特征构造，连接代数组合学与调和分析。
AI 中文摘要
会议信号是有限阿贝尔群上的复值函数，在时域和频域中均在0处取值为零，且在其他位置两个域中均为平坦（即模长恒定）。例如，勒让德符号是$\mathbb{Z}_p$上的会议信号，因为它在0处取值为零，在非零处取值为$\pm 1$，并且是其傅里叶变换的标量倍数。更一般地，有限域上的任何非平凡乘法特征都是其加法群上的会议信号。迄今为止，会议信号主要由数论学家研究，他们想知道乘法特征是否是有限域上仅有的会议信号。当前最先进的技术利用辛几何中的精巧工具证明了在非小域上存在非特征的会议信号。同样的工具还表明，在每个奇数阶阿贝尔群上都存在会议信号，但并未给出具体形式。在这篇（对读者友好的！）论文中，我们解释了会议信号如何产生代数组合学和应用调和分析中其他有趣的对象，然后我们给出了$\mathbb{Z}_n$上一些新的、显式的会议信号构造。特别地，我们首次给出了合数$n$的显式会议信号，以及素数$n$的显式非特征例子。
英文摘要
A conference signal is a complex-valued function on a finite abelian group that vanishes at 0 in both the time and frequency domains, and is otherwise flat in both domains. For example, the Legendre symbol is a conference signal on $\mathbb{Z}_p$ since it vanishes at $0$, takes values $\pm 1$ away from zero, and is a scalar multiple of its Fourier transform. More generally, any nontrivial multiplicative character on a finite field is a conference signal on its additive group. Until now, conference signals have mainly been studied by number theorists, who wanted to know if multiplicative characters were the only conference signals on finite fields. The state of the art uses fancy tools from symplectic geometry to prove that non-character conference signals exist on non-small fields. The same tools show conference signals exist on every abelian group of odd order, without saying what they are. In this (human-friendly!) paper, we explain how conference signals produce other objects of interest in algebraic combinatorics and applied harmonic analysis, and then we present some new, explicit constructions of conference signals on $\mathbb{Z}_n$. In particular, we give the first explicit conference signals for composite $n$ and the first explicit non-character examples for prime $n$.
