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

TL;DR

This paper compares the theory of fractal strings in real and p-adic contexts, highlighting their complex dimensions, volume formulas, and self-similarity properties, and discusses potential future research directions in nonarchimedean analysis.

Contribution

It provides a comprehensive survey of p-adic fractal strings, emphasizing their natural lattice structure and periodic complex dimensions, contrasting with the archimedean case.

Findings

p-adic self-similar strings are all lattice and have periodically distributed complex dimensions.

Explicit volume formulas for p-adic fractal strings are derived in terms of complex dimensions.

Comparison reveals key analogies and differences between real and p-adic fractal string theories.

Abstract

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.