The probability of the event $|S \cap T| = \infty$ is investigated, where $S$ is the trace of a random walk on the set of positive integers and $T$ is a fixed set of natural numbers.
@article{1176993918,
author = {Ruzsa, I. Z. and Szekely, G. J.},
title = {Intersections of Traces of Random Walks with Fixed Sets},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 132-136},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993918}
}
Ruzsa, I. Z.; Szekely, G. J. Intersections of Traces of Random Walks with Fixed Sets. Ann. Probab., Tome 10 (1982) no. 4, pp. 132-136. http://gdmltest.u-ga.fr/item/1176993918/