Almost sure recovery in quasi-periodic structures

The paper proves, under (H.1)–(H.2) and E|ξ|^{d+ε}<∞, that the random empirical Fourier averages M_{R,ξ,ω}(λ)dλ converge vaguely to ϕ(λ)·δ̂X, and leverages diffraction-theoretic tools (mutual singularity of diffraction measures, Lemma 4.1, Proposition 4.2) to treat i.i.d. and certain dependent perturbations; it also derives almost-sure recovery of δ̂X when ϕ≠0 (Theorem A, Cor. 2.1) . The candidate model gives a correct i.i.d.-case solution via a different route: test-function pairing, boundary-control by moment tails, and a Kolmogorov-type SLLN for the fluctuation term; it then constructs an explicit measurable recovery map when ϕ has no zeros. The only minor mismatch is that the model phrases (H.2) as a pure-point statement for δ̂X (sufficient for the paper’s arguments) while the paper formulates (H.2) via the diffraction measure and notes the pure-point Fourier transform setting as a sufficient instance; this does not affect correctness for the class of sets considered in the main theorem (which explicitly uses δ̂X in (2.1)) .