Let $0 < \delta_1 \leq \delta_2 \leq \delta_3 \leq 1$, and consider the box $B := [0,X^{\delta_1}] \times [0,X^{\delta_2}] \times [0,X^{\delta_3}] \subseteq \mathbb{R}^3$. Let $X > 3$ say. Is it possible to pack $B$ with cubes (box which has equal side lengths) $J_1, \ldots, J_N$ contained in $B$ satisfying the following property:

$(B \cap \mathbb{Z}^3) \subseteq \cup_{j=1}^N (J_i \cap \mathbb{Z}^3)$.

Given any $i \not = j$, $(J_i \cap \mathbb{Z}^3) \cap (J_j \cap \mathbb{Z}^3) = \emptyset.$

$N \leq (\log X)^{C} \frac{X^{\delta_1 + \delta_2 + \delta_3}}{X^{3 \delta_1}} $, where $C$ is a constant independent of $X$.

I would greatly appreciate any comments/references! Thank you very much!