  Quasiperiodic systems offer an appealing intermediate between long-range
ordered and genuine disordered systems, with unusual critical properties.
One-dimensional models that break the so-called self-dual symmetry usually
display a mobility edge, similarly as truly disordered systems in dimension
strictly higher than two. Here, we determine the critical localization
properties of single particles in shallow, one-dimensional, quasiperiodic
models and relate them to the fractal character of the energy spectrum. On the
one hand, we determine the mobility edge and show that it separates the
localized and extended phases, with no intermediate phase. On the other hand,
we determine the critical potential amplitude and find the universal critical
exponent $\nu \simeq 1/3$. We also study the spectral Hausdorff dimension and
show that it is nonuniversal but always smaller than unity, hence showing that
the spectrum is nowhere dense. Finally, applications to ongoing studies of
Anderson localization, Bose-glass physics, and many-body localization in
ultracold atoms are discussed.
