The ai-invariants of powers of ideals

Inspired by the recent work of Lu and O’Rourke, we study the ai-invariants of (symbolic) powers of some graded ideals. When I and J are two graded ideals in two distinct polynomial rings R and S over a common field K. We study the ai-invariants of the powers of the fiber product via the corresponding conditions on I and J. When IΔ is the Stanley-Reisner ideal of a k-dimensional complex Δ with k≥2. We investigate the ai-invariants of the symbolic powers of IΔ.