Quadratic lower bound for the minimum size of a no-unique-sums set

Establish whether there exists an absolute constant c>0 such that m(p)≥c(log p)^2 for every sufficiently large prime p, thereby determining whether the known upper bound implies m(p)=Θ((log p)^2).

Background

For an odd prime p, m(p) denotes the minimum cardinality of a subset A of the finite field with p elements such that every sum in A+A has at least two distinct representations as an unordered pair, with repeated elements allowed. The paper proves the lower bound m(p)≥2{-80}(log p/log log p)2 and an upper bound of order (log p)2, yielding m(p)=(log p){2+o(1)}.

The unresolved issue is whether the extraneous logarithmic-loss factor in the lower bound can be removed to obtain a bound of order (log p)2 with an absolute positive constant. The authors note that an affirmative answer, combined with their upper bound, would establish the sharper asymptotic order m(p)=Θ((log p)2).

References

Is there an absolute constant c>0 such that

m(p)\ge c(\log p)2

for every sufficiently large prime p? An affirmative answer, together with Theorem~\ref{thm:main-upper}, would give m(p)=\Theta((\log p)2). Corollary~\ref{cor:exponent} determines the logarithmic exponent but leaves this question open.

— A near-quadratic lower bound for sets with no unique sums  (2610.09349 - Hou et al., 7 Oct 2026) in Question 2.1 (labelled qu:quadratic), Section 2 / immediately after the proof of the lower bound