  Commensurability is of paramount importance in numerous strongly interacting
electronic systems. In the Fractional Quantum Hall effect, a rich cascade of
increasingly narrow plateaux appear at larger denominator filling fractions.
Rich commensurate structures also emerge, at certain filling fractions, in high
temperature superconductors and other electronic systems. A natural question
concerns the character of these and other electronic systems at irrational
filling fractions. Here we demonstrate that quasicrystalline structures
naturally emerge in these situations, and trigger behaviors not typically
expected of periodic systems. We first show that irrationally filled quantum
Hall systems cross over into quasiperiodically ordered configuration in the
thin-torus limit. Using known properties of quasicrystals, we argue that these
states are unstable against the effects of disorder, in agreement with the
existence of quantum Hall plateaux. We then study analogous physical situations
in a system of cold Rydberg atoms placed on an optical lattice. Such an
experimental setup is generally disorder free, and can therefore be used to
detect the emergent quasicrystals we predict. We discuss similar situations in
the Falicov-Kimball model, where known exact results can be used to establish
quasicrystalline structures in one and two dimensions. We briefly speculate on
possible relations between our theoretical findings and the existence of glassy
dynamics and other features of strongly correlated electronic systems.
