p-adic path set fractals and arithmetic

TL;DR

This paper studies a class of p-adic fractal sets generated by automata, demonstrating their closure under arithmetic operations and providing formulas for their Hausdorff dimensions, revealing unique p-adic properties.

Contribution

It introduces a new class of p-adic fractals characterized by automata and proves their closure under addition, multiplication, and Minkowski sum, with computable Hausdorff dimension formulas.

Findings

Closure under addition and multiplication by p-rational numbers

Minkowski sum of two sets remains in the class

Existence of computable Hausdorff dimension formulas

Abstract

This paper considers a class C(Z_p) of closed sets of the p-adic integers obtained by graph-directed constructions analogous to those of Mauldin and Williams over the real numbers. These sets are characterized as collections of those p-adic integers whose p-adic expansions are describeed by paths in the graph of a finite automaton issuing from a distinguished initial vertex. This paper shows that 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 set in the class is shown to also belong to this class. These results represent purely p-adic phenomena in that analogous closure properties do not hold over the real numbers. We also show the existence of computable formulas for the Hausdorff dimensions of such sets.