Erdős problem 785 · problems.science

Let $A,B\subseteq \mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\lvert A\cap [1,x]\rvert$ and similarly for $B(x)$ . Is it true that if $A(x)B(x)\sim x$ then $A(x)B(x)-x\to \infty$ as $x\to \infty$ ?

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10 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

785.lean

Retained formal statement7 of 10

Narkiewicz [Na59] proved that, under the given assumptions (and perhaps swapping $A$ and $B$ ) we must have $A(2x)/A(x)\to 1$ and $B(2x)/B(x)\to 2$ .

FormalConjectures/ErdosProblems/785.leanErdos785.erdos_785.variants.narkiewicz8 linesExact file

1∀ (A B : Set ℕ),2  A.Infinite →3    B.Infinite →4      Erdos785.IsExactAdditiveComplement A B →5        Filter.Tendsto (fun x => ↑(Erdos785.counting A (2 * x)) / ↑(Erdos785.counting A x)) Filter.atTop (nhds 1) ∧6            Filter.Tendsto (fun x => ↑(Erdos785.counting B (2 * x)) / ↑(Erdos785.counting B x)) Filter.atTop (nhds 2) ∨7          Filter.Tendsto (fun x => ↑(Erdos785.counting B (2 * x)) / ↑(Erdos785.counting B x)) Filter.atTop (nhds 1) ∧8            Filter.Tendsto (fun x => ↑(Erdos785.counting A (2 * x)) / ↑(Erdos785.counting A x)) Filter.atTop (nhds 2)

SolvedStatement only, no proof

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