Quantitative results of the Romanov type representation functions

Abstract: For $\alpha >0$, let $$\mathscr{A}={ a_1<a_2<a_3<\cdots\}$$ and $$\mathscr{L}=\{ \ell_1, \ell_2, \ell_3,\cdots\} \quad \text{(not~necessarily~different)}$$ be two sequences of positive integers with $\mathscr{A}(m)>(\log m)^\alpha $ for infinitely many positive integers $m$ and $\ell_m<0.9\log\log m$ for sufficiently integers $m$. Suppose further that $(\ell_i,a_i)=1$ for all $i$. For any $n$, let $f_{\mathscr{A},\mathscr{L}}(n)$ be the number of the available representations listed below $$\ell_in=p+a_i \quad \left(1\le i\le \mathscr{A}(n)\right),$$ where $p$ is a prime number. It is proved that $$\limsup_{n\to \infty } \frac{f_{\mathscr{A},\mathscr{L}}(n)}{\log\log n}>0,$$ which covers an old result of Erd\H os in 1950 by taking $a_i=2^i$ and $\ell_i=1$. One key ingredient in the argument is a technical lemma established here which illustrates how to pick out the admissible parts of an arbitrarily given set of distinct linear functions. The proof then reduces to the verifications of a hypothesis involving well--distributed sets introduced by Maynard, which of course would be the other key ingredient in the argument.