The Geometry of p-Adic Fractal Strings: A Comparative Survey

We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the tubular neighborhood of a p-adic fractal string Lp, expressed in terms of the underlying complex dimensions. Special attention will be focused on p-adic self-similar strings, in which the nonarchimedean theory takes a more natural form than its archimedean counterpart. In contrast with the archimedean setting, all p-adic self-similar strings are lattice and hence, their complex dimensions (as well as their zeros) are periodically distributed along finitely many vertical lines. The general theory is illustrated by some simple examples, the nonarchimedean Cantor, Euler, and Fibonacci strings. Throughout this comparative survey of the archimedean and nonarchimedean theories of fractal (and possibly, self-similar) strings, we discuss analogies and differences between the real and p-adic situations. We close this paper by proposing several directions for future research, including seemingly new and challenging problems in p-adic (or rather, nonarchimedean) harmonic and functional analysis, as well as spectral theory.