$p$-adic path set fractals and arithmetic

Abstract

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.