On the sumset of the primes and a linear recurrence (Q2855864)

\textit{N. P. Romanoff} [Math. Ann. 109, 668--678 (1934; JFM 60.0131.03)] showed that the set of integers representable as the sum of a prime and a power of two has positive lower asymptotic density, and \textit{K. S. Enoch Lee} [Int. J. Number Theory 6, No. 7, 1669--1676 (2010; Zbl 1242.11072)] established the same conclusion with the power of two replaced by a Fibonacci number. The authors adapt Romanoff's method to prove, more generally, that one may replace the power of two by an element of any integral non-degenerate linear recurring sequence \(u=(u_k)\) with separable characteristic polynomial. The most difficult part of the argument, relying on the fundamental theorem on \(S\)-units, involves establishing the convergence of the sum of the reciprocals of the so-called \(u\)-irregular primes. This, together with a bit of sieve theory, leads to the conclusion that \(\sum_{n \leqslant N} r(n)^2 \ll N\), where \(r(n)\) denotes the number of representations of \(n\) in the shape \(n=p+u_k\). A routine argument delivers the lower bound \(\sum_{n \leqslant N} r(n) \gg N\), and the desired theorem then follows immediately from the Cauchy-Schwarz inequality.