Model sets with precompact Borel windows

The paper’s Theorem 4.1 establishes items (a)–(k) for model sets with precompact Borel windows—BAP, density, autocorrelation γ = dens(L)·ω_{c(B)}, pure point diffraction with intensities |∫_B χ⋆|^2, existence of an ergodic invariant measure m on the extended hull with pure point dynamical spectrum, CPP, and measurability of eigenfunctions—via approximation of B by compact/open windows and known results for extremal-density weak/open model sets, together with results from [28,29] (pure point ⇔ BAP, genericity) and PSF tools (e.g., [40]) . The candidate’s solution proves the same statements using a distinct route: (i) generic points on the Kronecker torus via Lindenstrauss’ pointwise ergodic theorem for tempered Følner sequences; (ii) a direct periodization/Weil identity on T; (iii) a PSF for factorizable test functions; and (iv) a Besicovitch–Parseval argument to derive pure point diffraction and CPP. The only substantive mismatch is that the paper states the dynamical spectrum is generated by the Bragg positions, whereas the candidate asserts it is precisely the set of Bragg positions; the former is the correct and slightly weaker formulation (minor fix) .