Quantitative results of the Romanov type representation functions

Abstract

For α > 0, let$$\mathscr{A}=\{a_1 \lt a_2 \lt a_3\lt\cdots\}$$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)\gt(\log m)^\alpha $ for infinitely many positive integers m and $\ell_m\lt0.9\log\log m$ for sufficiently large 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$$\limsup_{n\to \infty } \frac{f_{\mathscr{A},\mathscr{L}}(n)}{\log\log n}\gt0,$$which covers an old result of Erdős 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.