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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FormalConjectures/ErdosProblems/

785.lean

Retained formal statement4 of 10

Chen and Fang [ChFa10] proved the stronger statement that $A(x)B(x)-x\to \infty$ if $A$ and $B$ are infinite sets such that $A+B$ contains all large integers and $\limsup_{x\to \infty}\frac{A(x)B(x)}{x}<\frac{5}{4}.$ They later [ChFa14] improved $5/4$ to $3-\sqrt{3}\approx 1.268$ .

FormalConjectures/ErdosProblems/785.leanErdos785.erdos_785.variants.chen_fang_limsup6 linesExact file

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

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