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 statement10 of 10

Sárközy and Szemerédi [SaSz94] proved that it is impossible for $A(x)B(x)-x=o(A(x)).$

FormalConjectures/ErdosProblems/785.leanErdos785.erdos_785.variants.sarkozy_szemeredi6 linesExact file

1∀ (A B : Set ℕ),2  A.Infinite →3    B.Infinite →4      Erdos785.IsExactAdditiveComplement A B →5        ¬(fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) =o[Filter.atTop] fun x =>6            ↑(Erdos785.counting A x)

SolvedStatement only, no proof

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