A PERIODICITY RESULT FOR TILINGS OF Z3 BY CLUSTERS OF PRIME-SQUARED CARDINALITY

The paper proves that if Z^3 admits a tiling by a finite set F with |F| = p^2 (p prime), then there exists an F-tiling that is a finite union of 1-periodic sets (Theorem 4.1) . It uses the dilation lemma to obtain the key spectral identity (4.2) and the set Z in (4.3), then shows the spectral measure is supported on a finite union of codimension-1 subtori ker χ_g (Lemma 3.10), and proceeds by a careful case analysis, invoking combinatorial “prism” lemmas when plane divisibility emerges, and handling the line-support regime via Appendix B to deduce weak 1-periodicity . The candidate solution outlines the same structure: ergodic/spectral setup; dilation lemma; spectral support on finitely many lines/planes; if plane-support yields combinatorial rigidity (prism) then periodicity; otherwise, line-support implies weak 1-periodicity via Bhattacharya/Greenfeld–Tao methods. This matches the paper’s argument and conclusions. One minor overstatement is the model’s claim that, in the prism/plane-support case, one can explicitly decompose the tiling into p sets each invariant in the prism axis; the paper instead establishes a 2-periodic tiling (hence 1-weak periodic) via Lemma 3.3, without the specific axis-invariance decomposition . Aside from this nuance, the reasoning tracks the paper closely, and both reach the same theorem.