Dense nonsumsets and cyclic-interval square roots

Definitions and mathematical statement

Let $p$ be an odd prime and regard $\mathbb F_p$ as an additive group. For $B\subseteq\mathbb F_p$, define $B+B=\{b_1+b_2:b_1,b_2\in B\}$, allowing $b_1=b_2$ and counting distinct sums only. Call $S\subseteq\mathbb F_p$ a double sumset if there exists some $B\subseteq\mathbb F_p$ with $S=B+B$ exactly. The set $B$ is not specified in advance and need not be a subset of $S$. The empty set is a double sumset by taking $B=\varnothing$.

Define \[ M(p)=\max\{|S|:S\subseteq\mathbb F_p, \text{no }B\subseteq\mathbb F_p\text{ has }B+B=S\}, \qquad D(p)=p-M(p). \] Determine the asymptotic size of $M(p)$, or more informatively of its deficit $D(p)$, as $p$ ranges over primes tending to infinity. Sharp bounds require both constructing large sets that fail to be double sumsets and proving that every set above a threshold is one. The equality requirement is crucial: containment of $S$ in a double sumset would be trivial by using the whole field. Likewise, representing $S$ as $B+C$ with two independently chosen sets is not the question. The optimization is over all subsets, with no symmetry, interval, multiplicative, or random-model restrictions. The source's historical bounds motivate the extremal function but are not taken as an assertion of the present best estimates.

Short English statement

How large can a subset of a prime field be while still failing to equal the sum of some set with itself?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001107 (AIM-COMBINATORICS-0232), 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 2.7. Original workshop question underlying AIM-COMBINATORICS-0232. https://aimath.org/WWN/additivecomb/additivecomb.pdf.
• S3 Noga Alon, Large sets in finite fields are sumsets. Extremal nonsumset context. https://web.math.princeton.edu/~nalon/PDFS/sumset.pdf.