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Statement

The Cantor string \(\mathcal{L}\) has lengths \(\ell_j\) occurring with multiplicities matching the middle-thirds construction, giving geometric zeta function

\[\zeta_{\mathcal L}(s)=\frac{3^{-s}}{1-2\cdot 3^{-s}}.\]

Its poles — the complex dimensions of the Cantor set — are

\[\omega_k=\frac{\log 2}{\log 3}+i\,\frac{2\pi k}{\log 3},\qquad k\in\mathbb{Z}.\]

The real part \(\operatorname{Re}\omega_k=\log 2/\log 3\) is the Minkowski (box) dimension; the nonzero imaginary parts, spaced by the period \(\mathbf{p}=2\pi/\log 3\), generate the log-periodic oscillations in the geometric counting function.

Assumptions

  1. Self-similar Cantor string built from the ratio-\(1/3\) middle-thirds construction, so the scaling ratio is \(r=1/3\) with \(2\) pieces.
  2. Complex dimensions are defined as the poles of the meromorphic continuation of \(\zeta_{\mathcal L}\) (Lapidus-van Frankenhuijsen framework).

Proof

  1. [verified] (Pole equation.) Poles occur where the denominator vanishes: \(1-2\cdot 3^{-s}=0\iff 3^{-s}=\tfrac12\iff -s\log 3=-\log 2 + 2\pi i k\), giving \(s=\dfrac{\log 2}{\log 3}+i\dfrac{2\pi k}{\log 3}\).
  2. [verified] (These are exactly zeros of the denominator.) Substituting \(\omega_k\), one gets \(3^{-\omega_k}=3^{-\log2/\log3}\cdot e^{-2\pi i k} =\tfrac12\cdot 1=\tfrac12\) for every integer \(k\), so \(1-2\cdot 3^{-\omega_k}=0\). SymPy simplifies \(3^{-\omega_k}\) to exactly \(1/2\).
  3. [asserted] (Interpretation of the pole data.) \(\operatorname{Re}\omega_k =\log2/\log3=0.6309297535714574\) equals the Minkowski/box dimension of the Cantor set. The imaginary parts \(2\pi k/\log 3\) are integer multiples of the oscillatory period \(\mathbf{p}=2\pi/\log 3=5.719201734760254\) (both numeric values machine-checked); their role as the source of the log-periodic (multiplicatively periodic) fluctuations in the volume of tubular neighborhoods is the interpretation supplied by the Lapidus-van Frankenhuijsen tube formula, asserted here rather than re-derived.

Verification

  • Symbolic (SymPy): simplify(3**(-omega_k)) returns exactly \(1/2\) for integer \(k\), and the denominator \(1-2\cdot 3^{-\omega_k}\) evaluates to \(0+0i\) at \(k=0,1,2\) (confirmed to machine precision).
  • Numerical: \(\operatorname{Re}\omega=\log2/\log3=0.6309297535714574\) matches the Cantor Minkowski dimension; oscillation period \(2\pi/\log3=5.719201734760254\) (both confirmed).
  • Asserted: the identification of \(\zeta_{\mathcal L}\)'s poles with the "complex dimensions" and their role in the tube-formula oscillation is the Lapidus-van Frankenhuijsen theory — the framework is cited; the pole locations within it are the machine-verified part.

Historical source

The theory of complex dimensions and geometric zeta functions was developed by Michel Lapidus and collaborators through the 1990s-2000s, consolidated in Lapidus & van Frankenhuijsen (2006).

Related concepts

Citations

  1. Lapidus, M. L., van Frankenhuijsen, M. Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. Springer (2006).
  2. Lapidus, M. L., Pomerance, C. "The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums." Proc. London Math. Soc. 66 (1993): 41-69.