Quasiperiodic Sets at Infinity and Meromorphic Extensions of Their Fractal Zeta Functions

Abstract

We construct an interesting family of unbounded sets at infinity having a quasiperiodic structure in their geometry. We study these sets by analyzing their complex dimensions—a generalization of the classical Minkowski dimension, which are defined analytically—as poles of a suitably defined Lapidus zeta function at infinity introduced in a previous work by the author. We define the tube zeta function at infinity and obtain a functional equation that relates it to the Lapidus zeta function at infinity much as in the classical setting of relative fractal drums. Furthermore, under suitable assumptions, we provide some general results about extending the fractal zeta functions at infinity beyond their abscissae of convergence. We also provide a sufficiency condition for Minkowski measurability as well as a bound from above for the upper Minkowski content directly from the corresponding zeta functions at infinity. We show that complex dimensions of quasiperiodic sets at infinity possess a quasiperiodic structure which can be either algebraic or transcendental. Furthermore, we use the quasiperiodic sets at infinity to construct a maximally hyperfractal set at infinity with prescribed Minkowski dimension, i.e., a set for which the abscissa of convergence of the associated fractal zeta function at infinity becomes its natural boundary.