Let $\epsilon > 0$. Is there some $r \ll_\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k \geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\epsilon$?

The set of integers of the form 2^k+p (where p is prime) has positive lower density.

Formalisation note: here we also allow p = 1 since this simplifies the code and is equivalent to the original statement.