  We study massive (reccurent) sets with respect to a certain random walk
$S_\alpha $ defined on the integer lattice $\mathbb{Z} ^d$, $d=1,2$. Our random
walk $S_\alpha $ is obtained from the simple random walk $S$ on $\mathbb{Z} ^d$
by the procedure of discrete subordination. $S_\alpha $ can be regarded as a
discrete space and time counterpart of the symmetric $\alpha $-stable L\'{e}vy
process in $\mathbb{R}^d$.
  In the case $d=1$ we show that some remarkable proper subsets of $\mathbb{Z}$
, e.g. the set $\mathcal{P}$ of primes, are massive whereas some proper subsets
of $\mathcal{P}$ such as Leitmann primes $\mathcal{P}_h$ are
massive/non-massive depending on the function $h$. Our results can be regarded
as an extension of the results of McKean (1961) about massiveness of the set of
primes for the simple random walk in $\mathbb{Z}^3$.
  In the case $d=2$ we study massiveness of thorns and their proper subsets.
