math.FA — Functional Analysis
COUNTABLY INFINITE-DIMENSIONAL SPACES OF CRYSTALLINE MEASURES
We construct a symmetric locally finite set Λ ⊂ R for which the space MΛ of crystalline measures with support and spectrum in Λ has countably infinite Hamel dimension; Meyer asked what the dimension of such spaces can be. More precisely, there is a sequence of crystalline measures such that every element of MΛ is a unique finite linear combination of them. Every layer of MΛ of fixed Hermite–Sobolev order is finite-dimensional, but the layer dimensions are unbounded; in one construction they grow linearly, and in general their growth can be prescribed up to a linear change of the order. (For every locally finite set the dimension is finite, countably infinite or equal to the continuum, and the countable case is exactly this behaviour of the layers.) The construction uses clusters of atoms that imitate derivatives of Dirac masses, which produce one-dimensional blocks of prescribed Hermite order. The main difficulty is to show that the union of infinitely many blocks supports no other crystalline measures. We prove this by placing the blocks at random scales, separated by rapidly growing gaps, and showing that every element of fixed Hermite order splits into finitely many block components. We also obtain disjoint sets A, B such that the crystalline measures with support in B and spectrum in A form a countably infinite-dimensional space, while MA = MB = {0}. The existence statement, without the symmetry of Λ, has been formalised and machine-checked in Lean 4; the formal development is recorded in Palomar