For $A\subseteq \mathbb{N}$, let the *upper density* of $A$ be defined by $$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
Let $$A = \{n\in\mathbb{N}: 2n \text{ is the sum of } 2 \text{ primes}\}.$$ Can it be shown that $\mu^+(A) > 0$? What about $\mu^+(A) \geq 1/2$?