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

Abstract

Let $Ω(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\ge$ 1:$Ω(m)$ $\le$ 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 $C_κ>0$, which is valid for all $K \ge 2$ and any fixed $κ>0$. This estimate controls the average arithmetic correlation among the shifts $a^a$ and allows the Romanoff argument to be carried out.

Disclosure

“□ Acknowledgments Huixi Li’s research is supported by the National Natural Science Foundation of China (Grant No. 12561001). The authors thank Liangxun Li for helpful discussion. ChatGPT was used as an auxiliary tool. All mathematical arguments, proofs, and computations were independently verified by the authors. References [1] Christian Ballot and Florian Luca. On the sumset”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-24T00:28:51.963Z

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