Farey sequence and Graham's conjectures

Abstract

Let $F_n$ be the Farey sequence of order $n$. For $S\subseteq F_n$ we let $\mathcal{Q}(S)=\left\{x/y:x,y\in S,x\le y\text{and}y\ne 0\right\}$. We show that if $\mathcal{Q}(S)\subseteq F_n$, then $|S|\le n+1$. 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 $S=\left\{0,1,\frac{1}{2},\dots ,\frac{1}{n}\right\}$ or $S=\left\{0,1,\frac{1}{n},\dots ,\frac{n-1}{n}\right\}$ 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.

arXiv comment6 pages