零调和密度蕴含零密度

中文摘要 AI 辅助

若一个整数集合的元素被视为紧群单位圆上连续字符的频率，且这些元素在测度论意义上足够稀疏，使得支撑在小区间开弧上的Radon测度的Fourier变换值能够插值任意全局Radon测度的值，则该集合具有零调和密度。相比之下，零密度刻画的是整数集合相对于实数轴上扩张区间的比例大小。尽管这两个概念分别源自抽象调和分析与初等数论，本文证明了零调和密度是零密度的充分条件，解决了文献中关于这两个概念关系的公开问题。证明利用了复分析中的经典间隙定理。

英文摘要

A set of integers has zero harmonic density if its elements, viewed as frequencies of continuous characters of the unit circle as a compact group, are measure-theoretically sparse enough that the values of Fourier transforms of Radon measures supported on small open arcs can interpolate those of any global Radon measure. In contrast, zero density quantifies the proportional size of a set of integers relative to expanding intervals on the real line. Although these notions arise from abstract harmonic analysis and elementary number theory, respectively, zero harmonic density is proven as a sufficient condition for zero density, resolving what has been stated in the literature as an open problem concerning the relation between these two notions. The proof makes use of classical gap theorems from complex analysis.

发表机构

• De La Salle University（德拉萨大学）

机构由 AI 辅助整理，请以论文原文为准。