HYPERUNIFORMITY AND NON-HYPERUNIFORMITY OF QUASICRYSTALS

The paper proves that in the 1×1 Euclidean cut‑and‑project setting with an interval window there exist examples where the centered diffraction η̂(Bε) decays to 0 more slowly than any power ε^α along a sequence of radii (Theorem 1.4), via an explicit lattice Γa and windows Wb and a quantitative lower bound derived from the diffraction formula (Corollary 4.1, Theorem 4.2). This directly contradicts the model’s asserted uniform upper bound η̂(Bε)=O(ε). The model’s argument makes a critical counting/estimation error when passing from blockwise hit-counts to an O(δ) bound; in particular, the replacement of the sum over blocks by 4∑j 1/(1+(jL)^2) and the claim that this is O(1/L) (hence O(ε)) omits the j=0 contribution and is false. The paper’s construction shows that for u_k=2m_k^{−γ} (γ>2) there is a single dual-lattice peak contributing ≳ m_k^{−2}, so η̂([−u_k,u_k])/u_k^δ diverges for any δ>2/γ, and for Liouville a for all δ>0, validating Theorem 1.4. See Theorem 1.4, Corollary 4.1, Theorem 4.2, and Lemma 4.3 for the precise argument.