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Erdős problem 32

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Does there exist a set A⊆NA \subseteq \mathbb{N} such that ∣A∩{1,…,N}∣=o((log⁡N)2)|A \cap \{1, \ldots, N\}| = o((\log N)^2) and every sufficiently large integer can be written as p+ap + a for some prime pp and a∈Aa \in A?

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5 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

32.lean

Retained formal statement2 of 5

Must every additive complement AA to the primes satisfy lim inf⁡N→∞∣A∩{1,…,N}∣log⁡N>1\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{\log N} > 1?

FormalConjectures/ErdosProblems/32.leanErdos32.erdos_32.variants.liminf_gt_one3 linesExact file
∀ (A : Set ℕ),  Erdos32.IsAdditiveComplementToPrimes A →    1 < Filter.liminf (fun N => ↑{x ∈ Finset.Icc 1 N | x ∈ A}.card / ↑(Real.log ↑N)) Filter.atTop
SolvedStatement only, no proof