Settling the Optimal Exponent Relating Sumsets and Difference Sets

Abstract

For a finite nonempty subset $A$ of an abelian group, let $\ddot{I}ƒ(A)=|A+A|/|A|$ and $\hat{I}´(A)=|A-A|/|A|$. The classical sum-difference inequalities state that $$\ddot{I}ƒ(A{)}^{1/2}\le \hat{I}´(A)\le \ddot{I}ƒ(A{)}^2.$$ 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 $$\frac{\log \ddot{I}ƒ(A_K)}{\log \hat{I}´(A_K)}⟶2,$$ 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.

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