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.
"Intersections of Traces of Random Walks with Fixed Sets." Ann. Probab. 10 (1) 132 - 136, February, 1982. https://doi.org/10.1214/aop/1176993918