Minkowski dimension and explicit tube formulas for $p$-adic fractal strings

TL;DR

This paper develops explicit tube formulas for $p$-adic fractal strings using complex dimensions, linking Minkowski dimension with zeta function convergence, and illustrates these with examples like the Cantor and Euler strings.

Contribution

It introduces a general fractal tube formula for $p$-adic strings and establishes the equivalence of Minkowski dimension with zeta function abscissa and growth rate, unifying real and $p$-adic cases.

Findings

Explicit volume formulas for $p$-adic fractal strings.

Minkowski dimension equals the zeta function's abscissa.

Unified explanation for complex dimensions in real and $p$-adic fractals.

Abstract

The local theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a $p$-adic fractal string $\mathcal{L}_p$, expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a $p$-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and $p$-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.