Applications of Distance and Tube Zeta Functions

Abstract

In this chapter, we show that some fundamental geometric and number-theoretic properties of fractals can be studied by using their distance and tube zeta functions. This will motivate us, in particular, to introduce several new classes of fractals. Especially interesting among them are the transcendentally quasiperiodic sets, since they can be placed at the crossroad between geometry and number theory. We shall need two deep results from transcendental number theory; namely, the theorem of Gel’fond–Schneider, and its extension due to Baker. In this context, the connecting link between the number theory and the geometry of fractals will be their tube zeta functions. A natural extension of the notion of distance zeta function leads us to introducing a general class of weighted zeta functions. Here, we introduce the space L ∞)(Ω): = ∩ p > 1L p (Ω), called the limit L ∞ -space, from which the weight functions are taken. Intuitively, a given weight function w from the space L ∞)(Ω) may only have very mild singularities, say, of logarithmic type. However, the set of singularities may be large, in the sense that its Hausdorff dimension can be arbitrarily close (and even equal) to N. A typical example is the function w(x) = logd(x, A) which appears under the integral sign when we differentiate the distance zeta function. We illustrate the efficiency of the use of distance zeta functions by computing the upper box dimension of several new classes of geometric objects, including geometric chirps, fractal nests and string chirps. These sets are closely related to bounded spirals and chirps in the plane. We also recall the construction of a class of fractals, called zigzagging fractals, for which the upper and lower box dimensions do not coincide, and show that the associated fractal zeta functions are alternating, in a suitable sense.