Hyperuniform Delone Realizations and Rigidity

We prove a measurable realization theorem for hyperuniform Delone point processes. In dimensions $d\ge 2$, for every prescribed $q\ge 1$, every essentially free ergodic p.m.p.\ action of ${\mathbb{R}}^d$ admits, at every sufficiently large prescribed intensity, a generating Delone realization whose return-time point process $\eta$ is measurably isomorphic to the original action and whose Bartlett spectrum ${\sigma }_{\eta }$ satisfies $${\sigma }_{\eta }(B_{\epsilon })=o({\epsilon }^{2q})(\epsilon \downarrow 0)\mathrm{.}$$ Thus arbitrarily high finite-order low-frequency suppression can be imposed without changing the prescribed measurable dynamics. The same realizations can be chosen with surface-order ball variance and linear rigidity to any prescribed finite order, while also being maximally rigid and almost surely bounded-displacement equivalent to a lattice. For essentially free Euclidean-motion actions whose translation subaction is ergodic, the construction can be made isotropic and $V$-ergodic, and hence $V$-weakly mixing. In dimension one, every essentially free ergodic flow admits generating Delone realizations with logarithmic interval discrepancy, maximal rigidity, and near-quadratic decay of the Bartlett spectrum at the origin.