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 $x\to \infty$ ?

Sources

Browse retained paths and inspect the exact material available for this Problem.

10 retained statements2415f78e850a

Open selected source

FormalConjectures/ErdosProblems/

785.lean

Retained formal statement3 of 10

Chen and Fang [ChFa15] proved $A(x)B(x)-x\ll A(x)^c$ cannot hold for any constant $c>0$ .

FormalConjectures/ErdosProblems/785.leanErdos785.erdos_785.variants.chen_fang8 linesExact file

1∀ (A B : Set ℕ),2  A.Infinite →3    B.Infinite →4      Erdos785.IsExactAdditiveComplement A B →5        ∀ (c : ℝ),6          0 < c →7            ¬(fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) =O[Filter.atTop] fun x =>8                ↑(Erdos785.counting A x) ^ c

SolvedStatement only, no proof

Sources

Search problems.science