  We study approach to the large-time jammed state of the deposited particles
in the model of random sequential adsorption. The convergence laws are usually
derived from the argument of Pomeau which includes the assumption of the
dominance, at large enough times, of small landing regions into each of which
only a single particle can be deposited without overlapping earlier deposited
particles and which, after a certain time are no longer created by depositions
in larger gaps. The second assumption has been that the size distribution of
gaps open for particle-center landing in this large-time small-gaps regime is
finite in the limit of zero gap size. We report numerical Monte Carlo studies
of a recently introduced model of random sequential adsorption on patterned
one-dimensional substrates that suggest that the second assumption must be
generalized. We argue that a region exists in the parameter space of the
studied model in which the gap-size distribution in the Pomeau large-time
regime actually linearly vanishes at zero gap sizes. In another region, the
distribution develops a threshold property, i.e., there are no small gaps below
a certain gap size. We discuss the implications of these findings for new
asymptotic power-law and exponential-modified-by-a-power-law convergences to
jamming in irreversible one-dimensional deposition.
