الصفحة 1
الصفحة 1
Fractal Geometry, Complex Dimensions and Zeta Functions : Geometry and Spectra of Fractal Strings
Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Key Features The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula The method of Diophantine approximation is used to study self-similar strings and flows Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions Throughout new results are examined. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts, and discusses several open problems and extensions.
Diophantine Approximation : Festschrift for Wolfgang Schmidt
This volume contains 22 research and survey papers on recent developments in the field of diophantine approximation. The first article by Hans Peter Schlickewei is devoted to the scientific work of Wolfgang Schmidt. Further contributions deal with the subspace theorem and its applications to diophantine equations and to the study of linear recurring sequences. The articles are either in the spirit of more classical diophantine analysis or of geometric or combinatorial flavor. In particular, estimates for the number of solutions of diophantine equations as well as results concerning congruences and polynomials are established. Furthermore, the volume contains transcendence results for special functions and contributions to metric diophantine approximation and to discrepancy theory.
