Strengthen the integer restricted-sumset bound for one-sided degree constraints

Prove that for finite subsets A,B of the integers with |B|≤|A| and a binary relation R⊆A×B whose maximum degree on B is at most D, the restricted sumset satisfies |A⧹+_R B|≥|A|+|B|−1−⌊5D/2⌋.

Background

The paper proves the lower bound |A⧹+_R B|≥|A|+|B|−3D when the relation has maximum degree at most D on the smaller side B. It then constructs examples attaining |A|+|B|−1−⌊5D/2⌋, suggesting that the proved −3D term is not optimal.

The authors identify the integer case as the bottleneck for improving the corresponding finite-field result, because their rectifiability argument reduces part of the finite-field analysis to subsets of the integers. The conjectured bound would therefore also strengthen the finite-field theorem with the same parameter dependence.

References

We believe the above theorem is tight, which suggests \Cref{thm:main_Z_case}(i) could potentially be strengthened to $|A \rplus B| |A| + |B| - 1 - \left\lfloor \frac{5\D}{2} \right\rfloor$.

— On restricted sumsets with bounded degree relations  (2503.09121 - Ouyang, 12 Mar 2025) in Section 1, immediately following Theorem 3 (the conjecture following Theorem 4)