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

Abstract

Let $\hat{I}\text{c◯}(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\ge$ 1:$\hat{I}\text{c◯}(m)$ $\le$ 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 \[ \frac{1}{K(K-1)} \sum_{\substack{1\le a,b\le K\\a\ne b}} \prod_{p\mid a^a-b^b}\left(1+\fracκ{p}\right) \le C_κ\] for some constant $C_{\hat{I}}º>0$, which is valid for all $K\ge 2$ and any fixed $\hat{I}º>0$. This estimate controls the average arithmetic correlation among the shifts $a^a$ and allows the Romanoff argument to be carried out.