For what functions $g (N) \to \infty$ is it true that $∣ A \cap {1, \dots, N}∣ ≫ \frac{N ^{1/2}}{g ( N )}$ implies $lim sup 1_{A} * 1_{A} (n) = \infty$ ?

unreviewedOpen. Worked here; no verified result yet.

number theory · open · prize $500 · formalized (Lean) · 0 attempts

machinery: additive-basis,representation-function-unboundedness,erdos-turan-conjecture,Sidon/B_h,additive-energy,square-root-density-threshold,sieve/Brun-Titchmarsh

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Evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (A\subseteq \mathbb N) and write [ r_A(n):=(1_A*1_A)(n)=|\\{(a,b)\in A^2:a+b=n\\}|. ]

candidate solution ↗

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

3 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.

candidate solution ↗

Formal proof

AMS 11 · open (literature)

theorem erdos_40 : Erdos40ForSet answer(sorry)

formal-conjectures/40.lean ↗

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