Minkowski Measurability and Exact Fractal Tube Formulas for p-Adic Self-Similar Strings

TL;DR

This paper develops an exact fractal tube formula for p-adic self-similar strings using complex dimensions, demonstrating non-Minkowski measurability and providing explicit formulas for average Minkowski content.

Contribution

It introduces a precise volume formula for p-adic fractal strings based on their complex dimensions, extending the theory of fractal geometry into the p-adic setting.

Findings

Lp is not Minkowski measurable.

Explicit formula for average Minkowski content.

Examples include 3-adic Cantor and 2-adic Fibonacci strings.

Abstract

The theory of p-adic fractal strings and their complex dimensions was developed by the first two authors in [17, 18, 19], particularly in the self-similar case, in parallel with its archimedean (or real) counterpart developed by the first and third author in [28]. Using the fractal tube formula obtained by the authors for p-adic fractal strings in [20], we present here an exact volume formula for the tubular neighborhood of a p-adic self-similar fractal string Lp, expressed in terms of the underlying complex dimensions. The periodic structure of the complex dimensions allows one to obtain a very concrete form for the resulting fractal tube formula. Moreover, we derive and use a truncated version of this fractal tube formula in order to show that Lp is not Minkowski measurable and obtain an explicit expression for its average Minkowski content. The general theory is illustrated by two simple examples, the 3-adic Cantor string and the 2-adic Fibonacci strings, which are nonarchimedean analogs (introduced in [17, 18]) of the real Cantor and Fibonacci strings studied in [28].