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Erdős problem 337

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Let A⊆NA\subseteq \mathbb{N} be an additive basis (of any finite order) such that ∣A∩{1,…,N}∣=o(N)\lvert A\cap \{1,\ldots,N\}\rvert=o(N). Is it true that lim⁡N→∞∣(A+A)∩{1,…,N}∣∣A∩{1,…,N}∣=∞? \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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4 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

337.lean

Retained formal statement2 of 4

This was generalised (to the replacement of A+AA+A by the hh-fold sumset hAhA for any h≥2h\geq 2) by Ruzsa and Turjányi [RT85].

FormalConjectures/ErdosProblems/337.leanErdos337.erdos_337.variants.h_fold6 linesExact file
∀ (h : ℕ),  2 ≤ h →    ∃ A,      A.IsAsymptoticAddBasis ∧        ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) ∧          ¬Filter.Tendsto (fun N => ↑(h • A ∩ Set.Icc 1 N).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop Filter.atTop
SolvedStatement only, no proof