Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions

Abstract

Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\dots$ be the Farey fractions of order $n$. We then prove that the inequality $(a_l-a_k)(b_l-b_k)\ge 0$ holds for all $k$ and $l>k$ with $l-k\le \left(\frac{1}{12}-o(1)\right)n$, sharpening an old result by Erdős. On the other hand, we will show that for all $n\ge 4$ there are $k,l$ with $k<l<k+\frac{n}{4}+5$ for which the product $(a_l-a_k)(b_l-b_k)$ is negative.

9 pages