Erdős problem 337 · problems.science

Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$ . Is it true that $\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?$

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FormalConjectures/ErdosProblems/

337.lean

Retained formal statement3 of 4

Ruzsa and Turjányi do prove (under the same hypotheses) that $\lim_{N\to \infty}\frac{\lvert (A+A+A)\cap \{1,\ldots,3N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty,$ and conjecture that the same should be true with $(A+A)\cap \{1,\ldots,2N\}$ in the numerator.

FormalConjectures/ErdosProblems/337.leanErdos337.erdos_337.variants.ruzsa_turjanyi5 linesExact file

1∀ (A : Set ℕ),2  A.IsAsymptoticAddBasis →3    ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) →4      Filter.Tendsto (fun N => ↑((A + A) ∩ Set.Icc 1 (2 * N)).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop5        Filter.atTop

OpenStatement only, no proof

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