2026/07/10 by Abhishek Khetan
#math.CO
We construct an exact cluster F⊆ℤ2 of cardinality 8 with full affine span, together with an F-tiling T, such that the orbit closure of T in \0,1\ℤ2 does not contain a 1-periodic F-tiling. Since every F-tiling is a low-complexity configuration with respect to the window F := \-a : a ∈ F\, this supplies a "non-degenerate" counterexample, in a strong sense, to Nivat's conjecture for non-convex windows. This answers, in the negative, a question of Kari and Moutot (2023) whether every such counterexample must be degenerate, in the sense that the probing window is contained in a coset of a proper finite-index sublattice. We complement this with a positive result: for every exact cluster F of full affine span whose cardinality is the square of a prime, every F-tiling has a 1-periodic F-tiling in its orbit closure. Together with Szegedy's theorem that every tiling by a cluster of prime cardinality is 1-periodic, this shows that no cluster of fewer than 8 cells can exhibit the phenomenon, with the possible exception of cardinality 6, which we leave open.