  We consider interacting Bose particles in an external local potential. It is
shown that large class of external quasicrystal potentials cannot sustain any
type of Bose-Einstein condensates. Accordingly, at spatial dimensions $D\leq 2$
in such quasicrystal potentials a supersolid is not possible via Bose-Einstein
condensates at finite temperatures. The latter also hold true for the
two-dimensional Fibonacci tiling. However, supersolids do arise at $D\leq 2$
via Bose-Einstein condensates from infinitely long-range, nonlocal
interparticle potentials.
