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Let's say that a subset $A$ of $\mathbb{N}$ has the Goldbach property if every even number $\geq4$ is the sum of two numbers of $A$.

Are there any results and examples of low density sets with these property or any results in general about bounds of the density of these sets?

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    $\begingroup$ Erdos showed that there are sets $S$ with density as low as $O(n^{1/2 + \epsilon})$ are actually additive bases; meaning every large positive integer lies in $2S$. For concrete sets, it should be accessible to prove that semi-primes of finite order (say, numbers with at most $k$ prime factors) have the Goldbach property, and they are not much denser than the primes themselves. $\endgroup$
    – Stanley Yao Xiao
    Commented Mar 10, 2015 at 17:35
  • $\begingroup$ I changed "spare" in the title to "sparse". Please roll back if this is not what you intended. $\endgroup$
    – Todd Trimble
    Commented Mar 10, 2015 at 18:24
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    $\begingroup$ See mathoverflow.net/questions/74594 $\endgroup$
    – David E Speyer
    Commented Mar 10, 2015 at 18:28

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