Title: Bass-note spectra for Euclidean geometries

Abstract:

We introduce the general problem of prescribing the spectrum, and in particular the bass-note spectrum, of an invariant differential (or Hecke) operator on a locally symmetric Riemannian space (building). We then focus on the Euclidean case of flat tori where the problems are closely connected to Mahler's study of lattice points in star bodies. In the case that the star body is "automorphic", homogeneous dynamics can be applied, and in some cases decisively.

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