In 1934, Romanoff proved that the following set has positive lower density:

$$\displaystyle \mathcal{R}(x)= \{n \in \mathbb{N} : n \leq x, n = p + 2^k \}$$

where $p$ is a prime and $k \geq 0$ is a non-negative integer. In 2010 Lee proved the analogous result when powers of 2 are replaced by terms in the Fibonacci sequence. Very recently, Ballot and Luca generalized the above to arbitrary non-degenerate linear recurrences (Ballot, Christian, Luca, Florian, $\textit{On the sumset of the primes and a linear recurrence}$, Acta Arithmetica 161.1 (2013)).

The basic idea behind the proof is to define $r(n)$ to be the number of ways to write $n$ as the sum of a prime and a term in the linear recurrence considered, and simultaneously obtain a lower bound for $\displaystyle \sum_{n \leq x} r(n) \gg x$ and and upper bound for $\displaystyle \sum_{n \leq x} r(n)^2 \ll x$, and then apply the Cauchy-Schwartz inequality to obtain the desired result.

Thus, the nature of the proof does not yield an explicit constant. So there are two questions that remain unanswered:

1) Can one obtain an asymptotic for $\#\mathcal{R}(x)$? Is the natural density expected to exist at all?

2) Can one obtain an explicit constant for the lower bound $\#\mathcal{R}(x) \gg x$?

Thanks for any insights. It would already be insightful to obtain an answer to the above two inquiries for the case of Romanoff's theorem.