Does there exist $B \subset N$ which is not an additive basis, but is such that for every set $A \subseteq N$ of Schnirelmann density $α$ and every $N$ there exists $b \in B$ such that $∣(A \cup (A + b)) \cap {1, \dots, N}∣ \geq (α + f (α)) N$ where $f (α) > 0$ for $0 < α < 1$ ?The Schnirelmann density is defined by $d_{s} (A) = N \geq 1 in f \frac{∣ A \cap { 1 , \dots , N }∣}{N} .$

Worked, still open.

number theory · solved · formalized (Lean) · 0 attempts

machinery: additive-combinatorics,Schnirelmann-density,additive-basis,probabilistic-construction,random-set

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

This is (essentially verbatim) **Erdős Problem #38**, and **it is currently open**: no example of such a set $B$ is known, and no proof that none can exist is known either.

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

AMS 11 · solved (literature)

theorem erdos_38 : answer(True) ↔
    ∃ B : Set ℕ, ¬ B.IsWeakAddBasis ∧ ∃ f : ℝ → ℝ, (∀ α, 0 < α → α < 1 → f α > 0) ∧
      ∀ (A : Set ℕ) (N : ℕ),
        let α := schnirelmannDensity A
        ∃ b ∈ B, (Ioc 0 N ∩ (A ∪ (A + {b}))).ncard ≥ (α + f α) * N

formal-conjectures/38.lean ↗

status

solved

evidence

formal

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