  Motivated by a recent experimental observation of superconductivity in the
Al-Zn-Mg quasicrystal, we study the low-temperature behavior of electrons
moving in the quasiperiodic potential of the Ammann-Beenker tiling in the
presence of a local attraction. We employ the Bogoliubov-de Gennes approach for
approximants of different sizes and determine the local pairing amplitude
$\Delta_{i}$ as well its spatial average, $\Delta_{0}$, the superconducting
order parameter. Due to the lack of periodicity of the octagonal tiling, the
resulting superconducting state is inhomogeneous, but we find no evidence of
the superconductivity islands, as observed in disordered systems, with
$\Delta_{i}\rightarrow0$ at $T_{c}$ for all sites. In the weak-coupling regime,
we find that the superconducting order parameter depends appreciably on the
approximant size only if the Fermi energy sits at a pseudogap in the
noninteracting density of states, with $\Delta_{0}$ decreasing as the system
size increases. These results are in line with the experimental observations
for the Al-Zn-Mg quasicrystal, and they suggest that, despite their electronic
structure, quasicrystals are prone to display conventional BCS-like
superconductivity.
