On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

Dubon, Eric; Sepulcre, Juan Matías

doi:10.1080/10586458.2013.853630

Cited by 10 publications

(4 citation statements)

References 10 publications

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“…Remark 3.9. References on or related to fractal strings include Fal2,DemDenKo Ü,DemKo Ö Ü,DubSep,ElLapMacRo,Fal2,Fr,HamLap,HeLap,KeKom,KomPeWi,LapLéRo,LapLu, Pe,PeWi,Ra1,, with applications to (or motivations from) a variety of subjects, such as number theory, fractal geometry, dynamical systems, harmonic analysis, spectral theory and mathematical physics.…”

Section: Statement Of the Main Results And Applicationsmentioning

confidence: 99%

Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

Lapidus¹,

Radunović²,

Žubrinić³

2016

Preprint

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We survey recent advances dealing with Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimensions in terms of their complex dimensions, which are defined as the poles of their associated fractal zeta functions. Relative fractal drums represent a far-reaching generalization of bounded subsets of Euclidean spaces as well as of fractal strings studied extensively by the first author and his collaborators. In fact, the Minkowski measurability criterion discussed here is a generalization of the corresponding one obtained for fractal strings by the first author and M. van Frankenhuijsen. Similarly as in the case of fractal strings, the criterion is formulated in terms of the locations of the principal complex dimensions associated with the relative drum under consideration. These complex dimensions are defined as poles or, more generally, singularities of the corresponding distance (or tube) zeta function associated. We also discuss the notion of gauge-Minkowski measurability of RFDs and survey several results connecting it to the nature and location of the complex dimensions. (This is especially useful when the underlying scaling law does not follow a classic power law.) We illustrate our results and their applications by means of a number of interesting examples.

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Section: Statement Of the Main Results And Applicationsmentioning

confidence: 99%

Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

Lapidus¹,

Radunović²,

Žubrinić³

2016

Preprint

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“…The result in (2.3) establishes a deep connection between the study of complex dimensions of self-similar fractal strings and that of the roots of Dirichlet polynomials which was gaining interest as early as the start of the nineteenth century; see, e.g., [21][22][23] and [8,26], along with the relevant references therein.…”

Section: Preliminary Materialsmentioning

confidence: 91%

“…A Dirichlet polynomial f is called lattice if w j /w 1 is rational for 1 ≤ j ≤ N , and it is called nonlattice otherwise. 8 It is straightforward to check that a Dirichlet polynomial f is lattice if and only if there exists a (necessarily unique) real number r in (0, 1), called the multiplicative generator of f , and positive integers k 1 , . .…”

Section: 2mentioning

confidence: 99%

Quasiperiodic patterns of the complex dimensions of nonlattice self-similar strings, via the LLL algorithm

Lapidus¹,

Frankenhuijsen²,

Voskanian³

2020

Preprint

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The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of the quasiperiodic patterns of the sets of complex dimensions of nonlattice self-similar fractal strings. The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A.

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“…The special case when N = 1 corresponds to bounded fractal strings, for which we must have that D ∈ [0, 1]; see, for example, HeLap,. Other references related to fractal strings include [DubSep,ElLapMacRo,Fal2,Fr,HamLap,Kom,LapLu,LapRo,LapLéRo,LapRoŽu,LéMen,MorSep,RatWi,.…”

Section: Introductionmentioning

confidence: 99%

Zeta functions and complex dimensions of relative fractal drums: theory, examples and applications

Lapidus

Radunović

Žubrinić

2017

Dissertationes Math.

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relative fractal drum (RFD), fractal zeta functions, relative distance zeta function, relative tube zeta function, geometric zeta function of a fractal string, relative Minkowski content, relative Minkowski measurability, relative upper box (or Minkowski) dimension, relative complex dimensions of an RFD, holomorphic and meromorphic functions, abscissa of absolute and meromorphic convergence, transcendentally ∞-quasiperiodic function, transcendentally ∞-quasiperiodic RFD, a-string of higher order.

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On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

Cited by 10 publications

References 10 publications

Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

Quasiperiodic patterns of the complex dimensions of nonlattice self-similar strings, via the LLL algorithm

Zeta functions and complex dimensions of relative fractal drums: theory, examples and applications

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