  We theoretically investigate the appearance of spatially modulated
superconducting states in mesoscopic superconducting thin-wall cylinders in a
magnetic field at low temperatures. Quantization of the electron motion around
the circumference of the cylinder leads to a discontinuous evolution of the
spatial modulation of the superconducting order parameter along the transition
line T_c(H). We show that this discontinuity leads to the non-monotonic
behavior of the specific heat jump at the onset of superconductivity as a
function of temperature and field. We argue that this geometry provides an
excellent opportunity to directly and unambiguously detect distinctive
signatures of the Fulde-Ferrell-Larkin-Ovchinnikov modulation of the
superconducting order.
