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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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

337.lean

Retained formal statement2 of 4

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

FormalConjectures/ErdosProblems/337.leanErdos337.erdos_337.variants.h_fold6 linesExact file

1∀ (h : ℕ),2  2 ≤ h →3    ∃ A,4      A.IsAsymptoticAddBasis ∧5        ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) ∧6          ¬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

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