# $p$-adic path set fractals and arithmetic

### William C. Abram

Hillsdale College, USA### Jeffrey C. Lagarias

University of Michigan, Ann Arbor, USA

## 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.

## Cite this article

William C. Abram, Jeffrey C. Lagarias, $p$-adic path set fractals and arithmetic. J. Fractal Geom. 1 (2014), no. 1, pp. 45–81

DOI 10.4171/JFG/2