Quantitative results of the Romanov type representation functions

Abstract

For $\alpha >0$, let $$\mathcal{A}=\{a_1<a_2<a_3<\cdots \}$$ and $$\mathcal{L}=\{{\ell }_1,{\ell }_2,{\ell }_3,\cdots \}\text{(not necessarily different)}$$ be two sequences of positive integers with $\mathcal{A}(m)>(\log m{)}^{\alpha }$ 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 $f_{\mathcal{A},\mathcal{L}}(n)$ be the number of the available representations listed below $${\ell }_{i}n=p+a_i\left(1\le i\le \mathcal{A}(n)\right),$$ where $p$ is a prime number. It is proved that $$\underset{n\to \infty }{\mathrm{limsup}}\frac{f_{\mathcal{A},\mathcal{L}}(n)}{\log \log n}>0,$$ 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.