  Inspired by recent experimental studies of local magnetic moments interacting
with a metallic quasicrystal, we study the low-temperature fate of spins placed
in the two-dimensional Ammann-Beenker tiling. In the diluted local moment
limit, we calculate the spin relaxation rate $1/T_{1}$, as measured by electron
spin resonance, and show that it displays a marked dependence on the system
size $N$ and the Fermi energy of the electronic bath. For a finite
concentration of spins, we integrate out the conduction electrons and generate
an effective magnetic coupling between the local moments, which we treat as
Ising spins. Despite the strongly frustrating nature of the magnetic couplings
and the lack of periodicity in the problem, we find long-range orders for
finite $N$ in our large-scale Monte Carlo simulations. However, the resulting
magnetically state is fragile, as clusters of essentially free spins fluctuate
down to very low temperatures.
