-adic path set fractals and arithmetic

  • William C. Abram

    Hillsdale College, USA
  • Jeffrey C. Lagarias

    University of Michigan, Ann Arbor, USA


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, -adic path set fractals and arithmetic. J. Fractal Geom. 1 (2014), no. 1, pp. 45–81

DOI 10.4171/JFG/2