Let $r_1,\dots ,r_t$ be positive integers and let $R_2(r_1,\dots ,r_t)$ be the set of positive odd integers that can be represented as $p+2^{k_1^{r_1}}+\cdots +2^{k_t^{r_t}}$, where p is a prime and $k_1,\dots ,k_t$ are positive integers. It is easy to see that if $r_1^{-1}+\cdots +r_t^{-1}<1$, then the set $R_2(r_1,\dots ,r_t)$ has asymptotic density zero. In this paper, we prove that if $r_1^{-1}+\cdots +r_t^{-1}\ge 1$, then the set $R_2(r_1,\dots ,r_t)$ has a positive lower asymptotic density. Several conjectures and open problems are posed for further research.

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关于 p+2k1r1+⋯+2ktrt 形式的整数

让$r_1,\dots \dots ,r_t$是正整数并且让${右}_2（r_1,\dots \dots ,r_t）$是正奇数的集合，可以表示为$p+2^{k_1^{r_1}}+\cdots +2^{k_t^{r_t}}$，其中p是素数并且$k_1,\dots \dots ,k_t$是正整数。很容易看出如果$r_1^{-1}+\cdots +r_t^{-1}<1$，那么集合${右}_2（r_1,\dots \dots ,r_t）$渐近密度为零。在本文中，我们证明如果$r_1^{-1}+\cdots +r_t^{-1}\ge 1$，那么集合${右}_2（r_1,\dots \dots ,r_t）$具有正的较低渐近密度。提出了一些猜想和未解决的问题以供进一步研究。