Self-similar p-adic fractal strings and their complex dimensions

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings. The text was submitted by the authors in English.