Settling the Optimal Exponent Relating Sumsets and Difference Sets

For a finite nonempty subset $A$ of an abelian group, let $\sigma(A)=|A+A|/|A|$ and $\delta(A)=|A-A|/|A|$. The classical sum-difference inequalities state that $$\sigma(A)^{1/2}\leq\delta(A)\leq\sigma(A)^2.$$ 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 $$\frac{\log\sigma(A_K)}{\log\delta(A_K)}\longrightarrow 2,$$ 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.