Large Sets of Integers with No Harmonic Triples

Abstract

Let $f(N)$ denote the largest size of a set $A\subseteq [N]=\{1,\ldots,N\}$ containing no distinct $a,b,c$ such that \[ \frac2a=\frac1b+\frac1c . \] We prove \[ f(N)\gg N\exp\!\left(-(2\sqrt{\log(24/7)}+o(1))\sqrt{\log\log N}\right). \] The construction filters the odd integers up to $N$ by a random affine image of a dense three-term-progression-free set in a prime field $\mathbb{F}_q$ with $q\asymp\log N$, and then deletes a controlled family of collapsed triples.