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

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.

34 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1105.2966 This is the final version of an original research article on the Minkowski dimension and explicit tube formulas for $p$-adic fractal strings. It is appeared in the open access journal Fractal Fractional

References in corpus (5)

• p-Adic Mathematical Physics
• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

Cited by in corpus (3)

• Distance and tube zeta functions of fractals and arbitrary compact sets
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces