Farey sequence and Graham's conjectures

Abstract

Let ${F}_{n}$ be the Farey sequence of order $n$. For $S \subseteq {F}_n$ we let $\mathcal{Q}(S) = \left\{x/y:x,y\in S, x\le y \, \, \textrm{and} \, \, y\neq 0\right\}$. We show that if $\mathcal{Q}(S)\subseteq F_n$, then $|S|\leq n+1$. Moreover, we prove that in any of the following cases: (1) $\mathcal{Q}(S)=F_n$; (2) $\mathcal{Q}(S)\subseteq F_n$ and $|S|=n+1$, we must have $S = \left\{0,1,\frac{1}{2},\ldots,\frac{1}{n}\right\}$ or $S=\left\{0,1,\frac{1}{n},\ldots,\frac{n-1}{n}\right\}$ except for $n=4$, where we have an additional set $\{0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}\}$ for the second case. Our results are based on Graham's GCD conjectures, which have been proved by Balasubramanian and Soundararajan.