Additive problem with $k$ numbers of a special form

In this paper, we consider an additive problem of the form $n_1+n_2+\ldots+n_k=N$ with at least two summands, where the summands satisfy the condition $n_i\in\mathbb{N}(\alpha_i, I_i)$ for $1 \le i \le k$, where $\mathbb{N}(\alpha I)=\{n\in\mathbb{N}:\{n\alpha\}\in I\}$.