GEOMETRIC AND O-MINIMAL LITTLEWOOD-OFFORD PROBLEMS

• 成果类型: Article
• 署名作者: Fox, Jacob; Kwan, Matthew; Spink, Hunter
• 署名单位: Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1590
• 发表日期: 2023
• 页码: 101-126
• 关键词: rational-points; POLYNOMIALS; singularity; THEOREMS; number; field

摘要

The classical Erdos-Littlewood-Offord theorem says that for nonzero vectors a1, ... , an E Rd, any x E Rd, and uniformly random (xi 1,. . . ,xi n) E {-1, 1}n, we have Pr(a1 xi 1 + middot middot middot + an xi n = x) = O(n-1/2). In this paper, we show that Pr(a1 xi 1 + middot middot middot + an xi n E S) < n-1/2+o(1) whenever S is definable with respect to an o-minimal structure (e.g., this holds when S is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.