Large Sets of Integers with No Harmonic Triples

Let $f(N)$ denote the largest size of a set $A\subseteq [N]=\{1,\dots ,N\}$ containing no distinct $a,b,c$ such that $$\frac{2}{a}=\frac{1}{b}+\frac{1}{c}\mathrm{.}$$ We prove $$f(N)\gg N\exp \text{\hspace{0.17em}⁣}(-(2\sqrt{\log (24\mathrm{/}7)}+o(1))\sqrt{\log \log N})\mathrm{.}$$ 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.