A generalization of the Romanoff theorem

Abstract

Let $\mathcal{P}$ be the set of primes and $\mathbb{N}$ the set of positive integers. Let also $r_1,...,r_t$ be positive real numbers and $R_2(r_1,...,r_t)$ the set of odd integers which can be represented as $$ p+2^{\lfloor k_1^{r_1}\rfloor}+\cdot\cdot\cdot+2^{\lfloor k_t^{r_t}\rfloor}, $$ where $p\in \mathcal{P}$ and $k_1,...,k_t\in\mathbb{N}$. Recently, Chen and Xu proved that the set $R_2(r_1,...,r_t)$ has positive lower asymptotic density, provided that $r_1^{-1}+\cdot\cdot\cdot+r_t^{-1}\ge 1$ and at least one of $r_1,...,r_t$ is an integer. Their result reduces to the famous theorem of Romanoff by taking $t=r_1=1.$ In this note, we remove the unnecessary condition that `{\it at least one of $r_1,...,r_t$ is an integer}'.