Measure Concentration on Fermi Balls

This note reports on some attempts to examine if and under which conditions the naturally scaled probability measures associated to an orthonormal basis of a classical Paley-Wiener space converge to a uniform distribution (on a compact set in momentum space). The results are still quite unsatisfactory, yet we got some indications that for inf-compact functions (symbols) and domains for which some generalized Weyl law holds, the measures converge weakly to a (generalized) Fermi ball.