Optimal exponent relating sumsets and difference sets

The problem asks whether the lower sum-difference inequality can hold with a uniformly stronger exponent. A 2026 construction shows that it cannot, even for finite subsets of the integers.

Known results

• Penman and Wells established only the lower bound $C\geq\frac{\log(32/5)}{\log(26/5)}=1.12594$ (2014).
• Staps proved the classical inequalities $\sigma(A)^{1/2}\leq\delta(A)\leq\sigma(A)^2$ and characterized equality: only cosets of finite subgroups (2014).

July 2026 construction

For every positive even integer $K$, the paper constructs explicit $A_K\subset\mathbb{Z}$ with $C(A_K)>\frac{2K}{K+3}\to2$. Hence $\sup_A C(A)=2$, equivalently the least universal exponent in $\sigma(A)\leq\delta(A)^c$ is $c=2$; the exponent $1/2$ is optimal. A Lean formalization is provided. The construction and proof were developed with assistance from Hyra, based on the open-weights Hy3 model.

Current status (as of July 2026): The exponent $1/2$ is settled as optimal, while the supremum is approached rather than attained for sets with at least two elements.