On integers of the form \(p+F_{2^k}+F_q\)

In 1934, Romanoff proved that the set of positive integers representable as the sum of a prime and a power of two has positive lower density.

Erdős later constructed an infinite arithmetic progression of odd integers none of which admits such a representation.

Let \(F_n\) be the Fibonacci sequence.

In this paper, we prove that the set of integers of the form \(p+F_{2^k}+F_q\), where \(p,q\) are primes and \(k\ge0\), has positive lower asymptotic density.

The same holds for the set of integers not of this form.