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.

Disclosure

“this work; Hyra with Hy3) sup CpAq “ 2 propose constructions and supporting arguments in natural language. Using GPT-5.6 Sol as the judge, we ran Hyra for approximately 24 hours, producing the construction underlying the paper. The LLM-based judgment was used only to guide exploration, not to certify mathematical correctness. We then independently checked the construc- tion and proof, corrected the exposition, and prepared the argument presented here manually. For con”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T19:12:10.133444Z

Structural counts