The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings

Lapidus, Michel L.; Maier, Helmut

doi:10.1112/jlms/52.1.15

Cited by 71 publications

(143 citation statements)

References 7 publications

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“…The question of whether fractals are Minkowski measurable has attracted recent prominence in work related to the Weyl-Berry conjecture on the distribution of eigenvalues of the Laplacian on domains with fractal boundaries; see [2,[8][9][10][11][12][13][14][15]. Particularly relevant is the paper by Lapidus and Pomerance [13] which gives an analysis of the conjecture in the 1-dimensional case.…”

Section: Final Remarksmentioning

confidence: 99%

On the Minkowski Measurability of Fractals

Falconer¹

1995

Proceedings of the American Mathematical Society

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Section: Final Remarksmentioning

confidence: 99%

On the Minkowski Measurability of Fractals

Falconer¹

1995

Proceedings of the American Mathematical Society

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“…We recall the notion of "fractal string" and "fractal spray from [30], see also [25,26,20,18,19,27,10] …”

Section: 1mentioning

confidence: 99%

Tube Formulas and Complex Dimensions of Self-Similar Tilings

Lapidus

Pearse

2010

Acta Appl Math

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Abstract. We use the self-similar tilings constructed in [32] to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubular zeta function and hence develop a tube formula for self-similar tilings in R d . The resulting power series in ε is a fractal extension of Steiner's classical tube formula for convex bodies K ⊆ R d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i = 0, 1, . . . , d − 1, just as Steiner's does. However, our formula also contains a term for each complex dimension. This provides further justification for the term "complex dimension". It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in [30].

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“…) We now show that the Minkowski dimension can be recovered as the abscissa of convergence of ζ L . (This was first observed by the first author in using a result of Besicovich and Taylor [BesT]; see also [LapPo1,LapMa].) Theorem 2.2.…”

Section: Fractal Stringsmentioning

confidence: 79%

“…This theory builds upon earlier work of the first author and his collaborators, including Carl Pomerance, Helmut Maier and Christina He 5,LapMa,HeLap]. In the present paper, we stress the important case of self-similar strings (which are one-dimensional self-similar geometries) along with the geometric and dynamical motivations and applications.…”

Section: Introductionmentioning

confidence: 99%

Fractality, self-similarity and complex dimensions

Lapidus

Frankenhuijsen

2004

Proceedings of Symposia in Pure Mathematics

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To Benoît Mandelbrot, on the occasion of his jubilee.Abstract. We present an overview of a theory of complex dimensions of selfsimilar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several strands to discuss a possible approach to establishing a cohomological interpretation of the complex dimensions.

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The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings

Cited by 71 publications

References 7 publications

On the Minkowski Measurability of Fractals

On the Minkowski Measurability of Fractals

Tube Formulas and Complex Dimensions of Self-Similar Tilings

Fractality, self-similarity and complex dimensions

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