  We study the statistics and the arithmetic properties of the Robin spectrum
of a rectangle. A number of results are obtained for the multiplicities in the
spectrum, depending on the Diophantine nature of the aspect ratio. In
particular, it is shown that for the square, unlike the case of Neumann
eigenvalues where there are unbounded multiplicities of arithmetic origin,
there are no multiplicities in the Robin spectrum for sufficiently small (but
nonzero) Robin parameter except a systematic symmetry. In addition, uniform
lower and upper bounds are established for the Robin-Neumann gaps in terms of
their limiting mean spacing. Finally, that the pair correlation function of the
Robin spectrum on a Diophantine rectangle is shown to be Poissonian.
