Spectrality and tiling by cylindric domains

A bounded set O in R^d is called spectral if the space L^2(O) admits an orthogonal basis consisting of
exponential functions. In 1974 Fuglede conjectured that spectral sets can be characterized geometrically
by their ability to tile the space by translations. Although since then spectral sets have been intensively
studied, the connection between spectrality and tiling is still unresolved in many aspects.
I will focus on cylindric sets and discuss a new result, joint with Nir Lev, on the spectrality of such sets.
Since also the tiling analogue of the result holds, it provides a further evidence of the strong connection
between these two properties.