Containments of symbolic powers of ideals of generic points in ${\text{PP}}^3$

We show that the Conjecture of Harbourne and Huneke, $I^{(Nr-(N-1))} \subset M^{(r-1)(N-1)}I^{r}$holdsforidealsofgeneric(simple)pointsin$\PP^3$. As a result, for such ideals we prove the following bounds, which can be recognized as generalizations of Chudnovsky bounds: $\alpha(I^{(3m-k)}) \geq m\alpha(I)+2m-k$,forany$m \geq 1$and$k=0,1,2$. Moreover, we obtain lower bounds for the Waldshmidt constant for such ideals.