Equal sums in random sets and the concentration of divisors

• 成果类型: Article
• 署名作者: Ford, Kevin; Ben Green; Koukoulopoulos, Dimitris
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign; University of Oxford; Universite de Montreal
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01177-y
• 发表日期: 2023
• 页码: 1027-1160
• 关键词: average orders

摘要

We study the extent to which divisors of a typical integer n are concentrated. In particular, defining Delta(n) := max(t) #{d vertical bar n, log d is an element of [t, t + 1]}, we show that Delta(n) >= (log log n)(0.35332277...) for almost all n, a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set A subset of N by selecting i to lie in A with probability 1/i. What is the supremum of all exponents beta(k) such that, almost surely as D -> infinity, some integer is the sum of elements of A boolean AND n [D-beta k, D] in k different ways? We characterise beta(k) as the solution to a certain optimisation problem over measures on the discrete cube {0, 1}(k), and obtain lower bounds for beta(k) which we believe to be asymptotically sharp.