Arithmetic purity of strong approximation for cubic hypersurfaces of large dimension

In this paper, we establish the arithmetic purity of strong approximation for smooth geometrically integral cubic hypersurfaces $X\subset \mathbb{P}^n$ over number fields $k$, provided that $n\ge 323$. For $k=\mathbb{Q}$, the bound can be lowered to $n \ge 30$.