Demailly's Conjecture for general and very general points

We prove that at least ${({\displaystyle \frac{(1+\epsilon )2m}{N-1}}+1+\epsilon )}^N$, where $0⩽\epsilon <1$, many general points, satisfy Demailly’s conjecture. Previously, it was known to be true for at least $(2m+2{)}^N$ many general points in arxiv.org/abs/2009.05022. We also study Demailly’s conjecture for $m=3$ for ideal defining general and very general points.