OPENHARDnumber theoryadditive basis
Is there a set $A\subset\mathbb{N}$ with $|A\cap\{1,\ldots,N\}| = O(\log N)$ such that every large integer can be written as $p+a$ for some prime $p$ and $a\in A$?
Notes: Erdos proved existence with << (log N)^2. Ruzsa improved to << omega(N) * log N and proved the lower bound liminf |A cap {1,...,N}| / log N >= e^gamma ~ 1.781.
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