Does there exist $A\subset \mathbb{N}$ with lower density $>1/3$ such that $a+b\neq 2^k$ for any $a,b\in A$ and $k\geq 0$?

Does there exist $A\subset \mathbb{N}$ with lower density $>1/3$ such that $a+b\neq 2^k$ for any $a,b\in A$ and $k\geq 0$?

Müller [Mu11] settled this question in the affirmative: in fact one can take $A$ to be the set of all integers congruent to $3\cdot 2^i\pmod{2^{i+2}}$ for any $i\geq 0$, which has density $1/2$.