Abstract:We study the solutions in s of a "Dirichlet polynomial equation" m 1 r s 1 + • • • + m M r s M = 1. We distinguish two cases. In the lattice case, when rj = r k j are powers of a common base r, the equation corresponds to a polynomial equation, which is readily solved numerically by using a computer. In the nonlattice case, when some ratio log rj/ log r1, j ≥ 2, is irrational, we obtain information by approximating the equation by lattice equations of higher and higher degree. We show that the set of lattice e… Show more
“…In the literature regarding the 1-dimensional case [28], [30], [5], the terms "gaps" and "multiple gaps" have been used where we have used "generators".…”
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].
“…In the literature regarding the 1-dimensional case [28], [30], [5], the terms "gaps" and "multiple gaps" have been used where we have used "generators".…”
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].
“…In that case, the transcendental equation (3.11) can no longer be transformed into a polynomial equation. This type of equation has been studied in [13], and it may be possible to extract precise information in our setting from the results in that paper.…”
Section: Corollary 33 If a = 0 Then The Single Point Extremal Funcmentioning
Abstract. This paper is devoted to examining some extremal problems in the Fock space. We discuss the order and type of Fock space functions and pose an extremal problem for a zero-based subspace corresponding to a finite zero set. We examine the zeros of the extremal function and solve an extremal problem for non-vanishing functions.
“…The theory of fractal strings has been developed over the past years by M. Lapidus and co-workers in a series of papers, including [6,10,12,14,15,16,17,19,20,21,22,23,24,25,26,27,32]. See also the book [18].…”
In this paper, we generalize the zeta function for a fractal string (as in Lapidus and Frankenhuijsen 2006 Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (New York: Springer)) in several directions.We first modify the zeta function to be associated with a sequence of covers instead of the usual definition involving gap lengths. This modified zeta function allows us to define both a multifractal zeta function and a zeta function for higher-dimensional fractal sets. In the multifractal case, the critical exponents of the zeta function ζ(q, s) yield the usual multifractal spectrum of the measure. The presence of complex poles for ζ(q, s) indicates oscillations in the continuous partition function of the measure, and thus gives more refined information about the multifractal spectrum of a measure. In the case of a self-similar set in R n , the modified zeta function yields asymptotic information about both the 'box' counting function of the set and the n-dimensional volume of the -dilation of the set.
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