  By engineering laser-atom interactions, both Hall ribbons and Hall cylinders
as fundamental theoretical tools in condensed matter physics have recently been
synthesized in laboratories. Here, we show that turning a synthetic Hall ribbon
into a synthetic Hall cylinder could naturally lead to localization. Unlike a
Hall ribbon, a Hall cylinder hosts an intrinsic lattice, which arises due to
the periodic boundary condition in the azimuthal direction, in addition to the
external periodic potential imposed by extra lasers. When these two lattices
are incommensurate, localization may occur on a synthetic Hall cylinder. Near
the localization-delocalization transitions, the dependence of physical
observables on the axial magnetic flux allows us to tackle a fundamental
question of determining the accuracy of rational approximation of irrational
numbers. In the irrational limit, physical observables are no longer affected
by fluctuations of the axial flux.
