The geometry of sets of parameters of wave packet frames

We study wave packet systems WP(ψ,M); that is, countable collections of dilations, translations, and modulations of a single function ψ ∈ L2(R). The parameters of these unitary actions form a discrete subset M ⊂ R+ × R × R. We introduce analogues of the notion of Beurling density, adapted to the geometry of discrete subsets of R+ × R × R, and notions of lower and upper dimensions associated with these densities. Our goal is to describe completeness properties of wave packet systems via geometric properties of the sets of their parameters. In particular, we show necessary conditions for WP(ψ,M) to be a Bessel system, and we construct multiple examples of non-standard wave packet frames with prescribed dimensions.