We obtain large gap asymptotics in the Bessel point process, in the case
where we apply the operation of a piecewise constant thinning on $m$
consecutive intervals. This operation consists of removing each particle on the
$j$th interval with probability $s_{j} \in [0,1]$, $j = 1,...,m$. We consider
two different regimes of the parameters: 1) the case $s_{1} > 0$, and 2) $s_{1}
= 0$ (i.e. there is no thinning on the first interval). In both cases we assume
$s_{2},...,s_{m} > 0$. The particular case of $m=1$ and $s_{1}=0$ is known and
corresponds to the large gap asymptotics for the Tracy-Widom distribution at
the hard edge.