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In this chapter, we analyze the oscillations in the geometry and the spectrum of the simplest type of generalized self-similar fractal strings. The complex dimensions of these generalized Cantor strings form a vertical arithmetic sequence + ( ∈ ℤ). We construct such a generalized Cantor string for any real-valued choice of and positive . We also construct for each positive integer ⋀ the so-called truncated Cantor strings, which have a finite arithmetic progression + of complex dimensions, where is restricted by Λ < < Λ.

As we saw in Chapter 10, the complex dimensions of a generalized Cantor string form an arithmetic progression + , with 0 < < 1 and > 0. In this chapter, we use this fact to study arithmetic progressions of critical zeros of zeta functions.

In this chapter, we make several suggestions for the direction of future research related to, and naturally extending in various ways, the theory developed in this book. In several places, we also provide some additional background material that may be helpful to the reader.