On arithmetic progressions in non-periodic tilings

The paper proves the 1D dichotomy cleanly: if the expansion λ is irrational, no infinite arithmetic progression (AP) of any finite patch occurs; if λ is rational (hence an integer), then “infinite AP” ⇔ “limit periodic” ⇔ “pure discrete spectrum” (Theorem 5.1, with the equivalence via Theorems 4.7, 4.8, and 4.15). The irrational case proof uses a robust circle-rotation/density argument that forces a contradiction by tile overlap, and requires no unproven structural assumptions beyond primitivity and self-similarity . By contrast, the model’s Part A hinges on an unjustified claim that inflating an AP of a fixed patch P yields further APs of the same patch at steps λ^n x with an offset independent of k, then deduces density of O_P − O_P. This recognizability-based alignment is not generally guaranteed for arbitrary P, so the core contradiction is unsupported. The model’s Part B (rational case) has the right overall equivalence and uses standard constant-length substitution/Dekking arguments, but it contains a technical mis-ordering of quantifiers when ensuring each inflated patch sits inside a single supertile before invoking a coincidence. The paper’s approach to the rational case (via overlap-coincidence/Meyer and limit-periodicity) avoids this pitfall and is complete for the 1D setting .