The largest support to escape chaos in random multiplicative functions

• 成果类型: Article
• 署名作者: Xu, Max Wenqiang
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-8424; 1091-6490
• DOI: 10.1073/pnas.2618812123
• 发表日期: 2026-09-22
• 页码: e2618812123
• 关键词: number theory; mathematical physics; probability theory; chaos

摘要

Let f(n) be a Steinhaus random multiplicative function and let A subset of[1,N] be a finite set of integers. We show that convergence of |A|-1/2 & sum;n is an element of Af(n) to a standard complex normal distribution forces |A|=o(N). We prove that this zero-density condition is sharp: For most sets A of density rho, and hence for some such sets, a complex Gaussian limit holds whenever (1-rho)-1=o((loglogN)1/2). However, for positive density, the correct normalization is (1-rho)|A| rather than |A|. Thus, the additional factor 1-rho is nontrivial precisely when the density does not tend to zero.