Sharp exponent relating sumset and difference-set growth

For a finite integer set A with at least two elements, compare its sumset growth σ=|A+A|/|A| with its difference-set growth δ=|A−A|/|A|. Is the exponent 1/2 in √σ≤δ sharp?

Why it matters and what counts as progress

Why it matters: Sharpness determines the strongest universal power comparison between addition and subtraction growth.

What counts as progress: Integer sets with log σ/log δ approaching 2, or a stronger universal inequality lowering that supremum; approaching a limit need not mean one set attains it.

How the problem developed

29 Jul 2026 · Optimal exponent [1][2]
Haowei Lin and Shanda Li, with the construction developed with Hyra Showed that the supremum of log(|A+A|/|A|) divided by log(|A−A|/|A|), over finite integer sets, is 2 and is not attained by any one set.

AI contribution

Settling the Optimal Exponent Relating Sumsets and Difference Sets

Haowei Lin, Shanda Li

For finite integer sets A with at least two elements, Lin and Li construct families with log(|A+A|/|A|)/log(|A−A|/|A|) approaching 2. Together with Ruzsa’s upper bound, this proves that the exponent 1/2 in √σ(A)≤δ(A) is sharp.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean 4.32.1 and Mathlib, with native_decide for three finite base-39 certificates + human)

Human understandingFull