Abstract
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.
Highlights
From 1991 to 1993, Lapidus, as well as Lapidus and Pomerance established connections between complex dimensions and the theory of the Riemann zeta function by studying the connection between fractal strings and their spectra; see [1,2,3]
This definition applies to fractal strings (self-similar strings have infinitely many nonreal complex dimensions; see, e.g., Equation (2.37) in [8] (Theorem 2.16).), that correspond to the ν = 1 case, and to bounded subsets of Rν as well as, more generally, to relative fractal drums, which are natural higher-dimensional counterparts of fractal strings
The lattice string approximation (LSA) algorithm is based on the theory of Diophantine approximations, which deals with the approximation of real numbers by rational numbers
Summary
From 1991 to 1993, Lapidus (in the more general and higher-dimensional case of fractal drums), as well as Lapidus and Pomerance established connections between complex dimensions and the theory of the Riemann zeta function by studying the connection between fractal strings and their spectra; see [1,2,3]. These include examples previously studied in [5,6,7,8], which can be viewed in a new light by using our refined numerical approach, and new handpicked examples which are computationally easier to explore and for which interesting new phenomena arise. The mathematical experiments performed in the current paper, along with earlier work in [6,7,8,15], have led to new questions and open problems which are briefly discussed in the concluding comments section, namely, Section 5
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