  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.
