论文标题
关于缩小分数的近似值的度量理论：定量的koukoulopoulos-maynard定理
On the metric theory of approximations by reduced fractions: a quantitative Koukoulopoulos-Maynard theorem
论文作者
论文摘要
令$ψ：\ mathbb {n} \ to [0,1/2] $。 Duffin-Schaeffer的猜想最近由Koukoulopoulos和Maynard解决，他断言，对于几乎所有的REALS $α$，都有无限的许多企业解决方案$（p，q）$ to不平等$ |α-p/q | <ψ（q） / q $，前提是系列$ \ sum_ {q = 1}^\inftyφ（q）ψ（q） / q $是不同的。在本文中，我们通过表明几乎所有$α$ coprime solutions $（P，Q）$的数量（均为$ Q \ leq Q $）为渐近订单$ \ sum_ {q = 1}^q2φ（q） / q $。证明依赖于Koukoulopoulos和Maynard发明的GCD图的方法，以及来自筛子理论的精致重叠估计，以及对“整数解剖学”的数字理论输入。关键现象是，随着设定系统的总质量的增加，近似组的系统平均表现出“平均渐近独立性”。
Let $ψ: \mathbb{N} \to [0,1/2]$ be given. The Duffin-Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $α$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|α- p/q| < ψ(q)/q$, provided that the series $\sum_{q=1}^\infty φ(q) ψ(q) / q$ is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all $α$ the number of coprime solutions $(p,q)$, subject to $q \leq Q$, is of asymptotic order $\sum_{q=1}^Q 2 φ(q) ψ(q) / q$. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate coming from sieve theory, and number-theoretic input on the "anatomy of integers". The key phenomenon is that the system of approximation sets exhibits "asymptotic independence on average" as the total mass of the set system increases.
