Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm
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 pro...
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MDPI AG
2021-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/6/591 |
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| author | Michel L. Lapidus Machiel van Frankenhuijsen Edward K. Voskanian |
| author_facet | Michel L. Lapidus Machiel van Frankenhuijsen Edward K. Voskanian |
| author_sort | Michel L. Lapidus |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T13:22:29Z |
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| language | English |
| last_indexed | 2024-03-10T13:22:29Z |
| publishDate | 2021-03-01 |
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| spelling | doaj.art-95b9e730813549c1a2f3a61051e5d8762023-11-21T09:54:28ZengMDPI AGMathematics2227-73902021-03-019659110.3390/math9060591Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL AlgorithmMichel L. Lapidus0Machiel van Frankenhuijsen1Edward K. Voskanian2Department of Mathematics, University of California, Riverside, CA 92521, USADepartment of Mathematics, Utah Valley University, Orem, UT 84058, USADepartment of Mathematics and Statistics, The College of New Jersey, Ewing, NJ 08618, USAThe 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.https://www.mdpi.com/2227-7390/9/6/591lattice and nonlattice self-similar stringsDiophantine approximationgeometric zeta functioncomplex dimensionsDirichlet polynomialroots of Dirichlet polynomials |
| spellingShingle | Michel L. Lapidus Machiel van Frankenhuijsen Edward K. Voskanian Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm Mathematics lattice and nonlattice self-similar strings Diophantine approximation geometric zeta function complex dimensions Dirichlet polynomial roots of Dirichlet polynomials |
| title | Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm |
| title_full | Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm |
| title_fullStr | Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm |
| title_full_unstemmed | Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm |
| title_short | Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm |
| title_sort | quasiperiodic patterns of the complex dimensions of nonlattice self similar strings via the lll algorithm |
| topic | lattice and nonlattice self-similar strings Diophantine approximation geometric zeta function complex dimensions Dirichlet polynomial roots of Dirichlet polynomials |
| url | https://www.mdpi.com/2227-7390/9/6/591 |
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