A negative answer to the Erdős-Sárkőzy question

Abstract

For a finite set $A$ of positive integers, let $H(A)$ be its set of subset sums, and let $g_3(n)$ be the least $N$ for which some $n$-element set $A\subseteq \{1,\dots ,N\}$ has $H(A)$ free of nonconstant three-term arithmetic progressions. Erdős and Sárkőzy asked whether $g_3(n)\gg 3^n$. We prove \[ \liminf_{n\to\infty}\frac{g_3(n)}{3^n}=0. \] More precisely, for every $\epsilon >0$ there is an integer $d\ge 2$ such that $g_3(d\ell )\le \epsilon 3^{d\ell }$ for all sufficiently large $\ell$. The proof uses Korsky's characterization of the problem in terms of ternary coefficient sums and a consequence of an OpenAI construction that provides positive integer coefficients whose linear form is injective on large integer boxes. A base-three expansion then gives the result.