A divisor-profile lower bound and a polynomial-height near-Sidon theorem for squares

Definitions and mathematical statement

Let $\mathcal Q=\{m^2:m\in\mathbb Z\}$ be the set of integer squares, including zero. For a positive integer $N$, consider a set $A\subseteq\mathcal Q$ of exactly $N$ distinct square values. Its sumset is \[ A+A=\{a+b:a,b\in A\}. \] Summands may coincide, and equal sums arising from different pairs are counted only once. There is no bound on the sizes of the squares in $A$ or on the integers whose squares they are. Define the extremal function \[ S(N)=\min_{\substack{A\subseteq\mathcal Q\\|A|=N}} |A+A|. \] The question is to obtain the strongest possible lower bound for $S(N)$ as a function of $N$, and ultimately determine its asymptotic growth by matching it with constructions.

The constraint is that every element individually is a square; the set need not consist of the first $N$ squares or any consecutive range of squares. In particular, estimating the sums of one such special family does not give a universal lower bound. One elementary normalization is $2N-1\le S(N)\le N(N+1)/2$: the first is the integer sumset bound and the second counts unordered pairs with repetition. The research target is the improvement forced by restricting to squares, including growth beyond any fixed multiple when applicable, rather than the number of pair representations of a typical sum. No probabilistic choice of $A$ is part of the assertion.

Short English statement

What is the sharp lower bound on the number of distinct pairwise sums of N distinct integer squares?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001119 (AIM-COMBINATORICS-0244), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Recent trends in additive combinatorics, item 3.7. Original workshop question underlying AIM-COMBINATORICS-0244. https://aimath.org/WWN/additivecomb/additivecomb.pdf.