Optimal Exponent Relating Sumsets and Difference Sets

For every finite set $A\subset\mathbb Z$ with $|A|\ge 2$, define $$C(A)=\frac{\log\left(|A+A|/|A|\right)} {\log\left(|A-A|/|A|\right)}.$$ Determine the largest possible value of $C(A)$, equivalently the least universal exponent $c$ such that $$\frac{|A+A|}{|A|} \le \left(\frac{|A-A|}{|A|}\right)^c$$ for every such set $A$. The result proves that the supremum is exactly $2$, although no individual admissible set attains it.

Result

Proved (New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.)

Status

Resolved

AI contribution

AI-discovered

Method

Argument

Field

Additive combinatorics

Posed by

—

Year posed

—

Years open

—

Solved

2026-07-29

Model

Hy3

Vendor

Tencent Hunyuan

Collaborators

Haowei Lin, Shanda Li

Verification

Lean-verified

Publication

Preprint

Significance

20 / 100*

Disclosed cost

—

Wikipedia

No dedicated article

Submitted by matthew