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

TL;DR

This paper presents an improved method for analyzing the quasiperiodic patterns of complex dimensions in nonlattice self-similar fractal strings using the LLL algorithm and advanced computational tools.

Contribution

It introduces a new, more powerful approach to studying the complex dimensions of nonlattice strings by combining the LSA algorithm with the LLL algorithm and multiprecision polynomial solving.

Findings

Enhanced visualization of quasiperiodic patterns

More accurate approximation of complex dimensions

Demonstrated effectiveness on various fractal strings

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. Lov\'asz.