AI 中文总结
研究Farey序列中顺序不佳的分数间Farey分数最小数量\(f(n)\)，通过证明\(f(n)=\left(\frac{1}{4}+o(1)\right)n\)得到匹配于Wouter van Doorn上界的下界，确定了相关渐近常数\(c = 1/4\)。
AI 中文摘要
设\(\mathcal{F}_n\)是按递增顺序排列的阶为\(n\)的Farey序列。若\(a < c\)且\(b > d\)，则称两个分数\(\frac{a}{b} < \frac{c}{d}\)顺序不佳。设\(f(n)\)是\(\mathcal{F}_n\)中两个顺序不佳的分数之间严格的Farey分数的最小数量。我们证明\(f(n)=\left(\frac{1}{4}+o(1)\right)n\)。在Erdős问题1005的等效索引约定中，这确定了所需的渐近常数\(c = 1/4\)。上界\(f(n)\le n/4+O(1)\)首先由Wouter van Doorn得到；这里的主要结果是匹配的下界。
英文摘要
Let $\mathcal{F}_n$ be the Farey sequence of order $n$, written in increasing order. Call two fractions $\frac{a}{b} < \frac{c}{d}$ badly ordered if $a < c$ and $b > d$. Let $f(n)$ be the minimum number of Farey fractions strictly between two badly ordered fractions in $\mathcal{F}_n$. We prove $f(n)=\left(\frac{1}{4}+o(1)\right)n$. In the equivalent indexing convention of Erdős Problem 1005, this determines the requested asymptotic constant as $c=1/4$. The upper bound $f(n)\le n/4+O(1)$ was first obtained by Wouter van Doorn; the main result here is the matching lower bound.
