Optimal result on restricted sumsets containing powers of two

It is proved that for all sufficiently large positive integers n, if $A\subseteq [1,n]$ with $\left|A\right|>\frac{n}{6}+2$ and $\gcd A=1$, then there exists a power of 2 which can be represented as the sum of at most 22 distinct elements of A. This answers a question in [13]. The result is optimal in two aspects. In the above conclusion, the condition $\left|A\right|>\frac{n}{6}+2$ cannot be replaced by $\left|A\right|>\frac{n}{6}+1$, and the number 22 is also best possible.