  We study bosons on the real line in a Poisson random potential (Luttinger--Sy
model) with contact interaction in the thermodynamic limit at absolute zero
temperature. We prove that generalized Bose--Einstein condensation (BEC) occurs
almost surely if the intensity $\nu_N$ of the Poisson potential satisfies $[\ln
(N)]^4/N^{1 - 2\eta} \ll \nu_N \lesssim 1$ for arbitrary $0 < \eta \leq 1/3$.
We also show that the contact interaction alters the type of condensation,
going from a type-I BEC to a type-III BEC as the strength of this interaction
is increased. Furthermore, for sufficiently strong contact interactions and $0
< \eta < 1/6$ we prove that the mean particle density in the largest interval
is almost surely bounded asymptotically by $\nu_NN^{3/5+\delta}$ for $\delta >
0$.
