This paper considers a class of subsets of the p -adic integers obtained by graph-directed constructions analogous to that of Mauldin and Williams over the real numbers. These sets are characterized as collections of p -adic integers whose p -adic expansions are described by paths in the graph of a finite automaton issuing from a distinguished initial vertex. This paper shows this class of sets is closed under the arithmetic operations of addition and multiplication by p -integral rational numbers. In addition the Minkowski sum (under p -adic addition) of two sets in the class belongs to the class. These results represent purely p -adic phenomena in that analogous closure properties do not hold over the real numbers. The paper also derives computable formulas for the Hausdorff dimensions of such sets.