  Polymer Quantization (PQ) is a background independent quantization scheme
that is deployed in Loop Quantum Gravity. This framework leads to a new
short-distance (discretized) structure characterized by a fundamental length.
In this paper we use PQ to analyze the problem of a particle bouncing on a
perfectly reflecting surface under the influence of Earth's gravitational
field, what we have called "\textit{The Polymer Bouncer}". In this scenario,
deviations from the usual quantum effects are induced by the spatial
discreteness, but not by a new short-range gravitational interaction. We solve
the polymer Schr\"odinger equation in an analytical fashion, and we evaluate
numerically the corresponding energy levels. We find that the polymer energy
spectrum exhibits a negative shift compared to the obtained for the quantum
bouncer. The comparison of our results with those obtained in the GRANIT
experiment leads to an upper bound for the fundamental length scale, namely
$\lambda \ll 0.6 \buildrel _{\circ} \over {\mathrm{A}}$. We find polymer
corrections to the probability of transitions between levels, induced by small
vibrations, together with the probability of spontaneous emission in the
quadrupole approximation.
