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 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.

Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.

Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis

This research expository article not only contains a survey of earlier work but also contains a main new result, which we first describe. Given c ≥0, the spectral operator can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding c . Rigorously, it turns out to coincide with a suitable quantization of the Riemann zeta function ζ = ζ ( s ): , where ∂=∂ c is the infinitesimal shift of the real line acting on the weighted Hilbert space . In this paper, we establish a new asymmetric criterion for the Riemann hypothesis (RH), expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter (i.e. for all c in the left half of the critical interval (0,1)). This corresponds (conditionally) to a mathematical (and perhaps also, physical) ‘phase transition’ occurring in the midfractal case when . Both the universality and the non-universality of ζ = ζ ( s ) in the right (resp., left) critical strip (resp., ) play a key role in this context. These new results are presented here. We also briefly discuss earlier joint work on the complex dimensions of fractal strings, and we survey earlier related work of the author with Maier and with Herichi, respectively, in which were established symmetric criteria for the RH, expressed, respectively, in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension D ∈(0,1), with , and of the quasi-invertibility of the family of spectral operators (with ).

EIGENVALUES FOR HIGH ORDER ELLIPTIC OPERATORS IN A FRACTAL STRING

In this paper, we study the spectrum of order 2m (m≥1) elliptic operator A in a bounded open set Ω∈R1, with fractal boundary Γ=∂Ω and Minkowski dimension D∈(0, 1), thus proving the corresponding modified Weyl-Berry conjecture to be true, namely [Formula: see text] where N(λ, A, Ω) is the counting function, [Formula: see text], C1, D =2-(1-D) π-D(1-D)(-ζ(D)), ζ(D) is the classical Riemann–zeta function, and ℳ(D, Γ) is the Minkowski measure of Γ.