A Romanoff-type theorem for $P_2$+{$a^a$: a$\ge$ 1}

Let $\mathrm{\Omega }(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\ge$ 1:$\mathrm{\Omega }(m)$ $\le$ 2}. We prove that the sumset $P_2$+{$a^a$: a$\ge$ 1} has positive lower density. The proof uses the Romanoff second moment method, in the spirit of Li and Pan’s theorem on $P_2$+$2^{\mathcal{P}}$. The main new ingredient is the following average estimate for the singular factor $$\frac{1}{K(K-1)}\sum _{\begin{array}{c}1\le a,b\le K\\ a\mathrm{\ne }b\end{array}}\prod _{p\mid a^a-b^b}(1+\frac{\kappa }{p})\le C_{\kappa }$$ for some constant $C_{\kappa }>0$, which is valid for all $K\ge 2$ and any fixed $\kappa >0$. This estimate controls the average arithmetic correlation among the shifts $a^a$ and allows the Romanoff argument to be carried out.