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Erdős problem 785

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

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10 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

785.lean

Retained formal statement9 of 10

Ruzsa [Ru17] has constructed, for any function w(x)→∞w(x)\to \infty, such a pair of sets with A(x)B(x)−x<w(x)A(x)B(x)-x<w(x) for infinitely many xx.

FormalConjectures/ErdosProblems/785.leanErdos785.erdos_785.variants.ruzsa_upper_bound7 linesExact file
∀ (w : ℕ → ℝ),  Filter.Tendsto w Filter.atTop Filter.atTop →    ∃ A B,      A.Infinite ∧        B.Infinite ∧          Erdos785.IsExactAdditiveComplement A B ∧            ∃ᶠ (x : ℕ) in Filter.atTop, ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x < w x
SolvedStatement only, no proof