NDLI: Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm

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Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm

Content Provider	MDPI
Author	Lapidus, Michel van Frankenhuijsen, Machiel Voskanian, Edward
Copyright Year	2021
Description	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. K. Lenstra, H. W. Lenstra, and L. Lovász.
Starting Page	591
e-ISSN	22277390
DOI	10.3390/math9060591
Journal	Mathematics
Issue Number	6
Volume Number	9
Language	English
Publisher	MDPI
Publisher Date	2021-03-10
Access Restriction	Open
Subject Keyword	Mathematics Mathematical Physics Lattice and Nonlattice Self-similar Strings Diophantine Approximation Geometric Zeta Function Complex Dimensions Dirichlet Polynomial Roots of Dirichlet Polynomials Lattice Case Nonlattice Case Lattice String Approximation (lsa) Algorithm Quasiperiodic Structure and Patterns Simultaneous Diophantine Approximation Lll Algorithm Algorithm of Lenstra, Lenstra and Lovász
Content Type	Text
Resource Type	Article

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