$p$-adic fractal strings of arbitrary rational dimensions and Cantor strings

Abstract: The local theory of complex dimensions for real and $p$-adic fractal strings describes oscillations that are intrinsic to the geometry, dynamics and spectrum of archimedean and nonarchimedean fractal strings. We aim to develop a global theory of complex dimensions for ad`elic fractal strings in order to reveal the oscillatory nature of ad`elic fractal strings and to understand the Riemann hypothesis in terms of the vibrations and resonances of fractal strings. We present a simple and natural construction of self-similar $p$-adic fractal strings of any rational dimension in the closed unit interval $[0,1]$. Moreover, as a first step towards a global theory of complex dimensions for ad`elic fractal strings, we construct an ad`elic Cantor string in the set of finite ad`eles $\mathbb{A}_0$ as an infinite Cartesian product of every $p$-adic Cantor string, as well as an ad`elic Cantor-Smith string in the ring of ad`eles $\mathbb{A}$ as a Cartesian product of the general Cantor string and the ad`elic Cantor string.