On the Duffin-Schaeffer conjecture

• 成果类型: Article
• 署名作者: Koukoulopoulos, Dimitris; Maynard, James
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.192.1.5
• 发表日期: 2020
• 页码: 251-307
• 关键词: theorems

摘要

Let psi : N -> R->= 0 be an arbitrary function from the positive integers to the non-negative reals. Consider the set A of real numbers a for which there are infinitely many reduced fractions a/q such that vertical bar alpha-a/q vertical bar <= psi(q)/q. If Sigma(infinity)(q=1) psi(q)phi(q)/q = infinity, we show that A has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality vertical bar alpha - a/q vertical bar <= psi(q)/q, giving a refinement of Khinchin's Theorem.