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 statement2 of 10

Chen conjectures that this should be true with $3/2$ .

FormalConjectures/ErdosProblems/785.leanErdos785.erdos_785.variants.chen_conjecture8 linesExact file

1True ↔2  ∀ (A B : Set ℕ),3    A.Infinite →4      B.Infinite →5        IsAdditiveComplement A B →6          Filter.limsup (fun x => ↑(↑(Erdos785.counting A x) * ↑(Erdos785.counting B x)) / ↑↑x) Filter.atTop <7              ↑(3 / 2) →8            Filter.Tendsto (fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) Filter.atTop Filter.atTop

OpenStatement only, no proof

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