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There are competing schools of thought about the question of whetherspacetime is fundamentally either continuous or discrete. Here, we consider thepossibility that spacetime could be simultaneously continuous and discrete, inthe same mathematical way that information can be simultaneously continuous anddiscrete. The equivalence of continuous and discrete information, which is ofkey importance in information theory, is established by Shannon samplingtheory: of any bandlimited signal it suffices to record discrete samples to beable to perfectly reconstruct it everywhere, if the samples are taken at a rateof at least twice the bandlimit. It is known that physical fields on genericcurved spaces obey a sampling theorem if they possess an ultraviolet cutoff.Most recently, methods of spectral geometry have been employed to show thatalso the very shape of a curved space (i.e., of a Riemannian manifold) can bediscretely sampled and then reconstructed up to the cutoff scale. Here, wedevelop these results further, and we here also consider the generalization tocurved spacetimes, i.e., to Lorentzian manifolds.

Spacetime could be simultaneously continuous and discrete in the same way that information can