Settling the Optimal Exponent Relating Sumsets and Difference Sets

Abstract

For a finite nonempty subset $A$ of an abelian group, let $σ(A)=|A+A|/|A|$ and $δ(A)=|A-A|/|A|$. The classical sum-difference inequalities state that $$σ(A)^{1/2}\leqδ(A)\leqσ(A)^2.$$ The exponent $2$ in the second inequality is known to be optimal, whereas it has remained open whether the exponent $1/2$ in the first inequality can be improved. We settle this question by constructing an explicit family of finite sets $A_K\subset\mathbb{Z}$ such that $$\frac{\logσ(A_K)}{\logδ(A_K)}\longrightarrow 2,$$ hence the exponent $1/2$ in the first inequality is also optimal. The construction and its proof were developed with the assistance of Hyra, an AI research agent based on the open-weights Hy3 model.