FRACTAL GEOMETRY AND NUMBER THEORY: COMPLEX DIMENSIONS OF FRACTAL STRINGS AND ZEROS OF ZETA FUNCTIONS

2001 ◽  
Vol 33 (2) ◽  
pp. 254-255
Author(s):  
L. Olsen
Keyword(s):  
Number Theory ◽  
Fractal Geometry ◽  
Zeta Functions ◽  
Fractal Strings ◽  
Complex Dimensions

scholarly journals Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

2018 ◽  
Vol 2 (4) ◽  
pp. 26 ◽  
Author(s):  
Michel Lapidus ◽  
Hùng Lũ’ ◽  
Machiel Frankenhuijsen
Keyword(s):  
Counting Function ◽  
Minkowski Dimension ◽  
Volume Formula ◽  
Fractal Strings ◽  
Complex Dimensions ◽  
Abscissa Of Convergence ◽  
Tubular Neighborhoods ◽  
Fractal String ◽  
Tube Formulas ◽  
Asymptotic Growth Rate

The 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 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.

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