On the expected number of distinct points in a subset visited by anN-step random walk

Many investigators have calculated asymptotically valid expressions for the expected number of distinct points visited by ann-step random walk on a lattice. In this note we point out that the same formalism can be used to study the expected number of distinct points in a subset of lattice points. We also calculate the expected occupancy of the subset and give sufficient conditions for the ratio of the two calculated quantities to have the same asymptotic time dependence as for the full lattice. Specific examples are considered.