OPENHARDnumber theoryadditive combinatorics
Does there exist $B\subset\mathbb{N}$ which is not an additive basis, but for every set $A$ of Schnirelmann density $\alpha$ and every $N$, there exists $b\in B$ such that $|(A\cup (A+b))\cap \{1,\ldots,N\}|\geq (\alpha+f(\alpha))N$ where $f(\alpha)>0$ for $0<\alpha <1$?
Notes: Linnik constructed an essential component that is not an additive basis. Erdos proved that if B is a basis of order k, the density increment is at least alpha(1-alpha)/(2k).
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