A COUNTEREXAMPLE TO THE PERIODIC TILING CONJECTURE

Rachel Greenfeld, Terence Tao

correcthigh confidence

Category: Not specified
Journal tier: Top Generalist
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the existence of a finite abelian 2-group G0 and a finite non-empty F ⊂ Z^2 × G0 such that A ⊕ F = Z^2 × G0 is aperiodic (all solutions exist but none are periodic), via an encoding of a 2-adic “Sudoku” with good columns into a single translational monotiling; this is Theorem 1.3 and its blueprint is laid out in Sections 1–9. The candidate solution follows the same Greenfeld–Tao blueprint and reaches the correct conclusion, though it omits the crucial “good columns” non-degeneracy and informally describes the Sudoku as a Z^2 SFT and invokes a Fourier-slice language that is not explicitly used in the paper. With these clarifications, the candidate’s outline is essentially the paper’s argument.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} top generalist

\textbf{Justification:}

This paper resolves a central conjecture in translational tiling by producing an aperiodic single-tile example in an abelian setting, and propagates the counterexample to Zd and Rd in high dimension. The technical contributions—especially the Sudoku-with-good-columns mechanism, the functional-equation tiling language, and the concatenation to a single tile—are original and broadly influential. While the manuscript is well organized, the densest sections (Sudoku encoding and aperiodicity) would benefit from additional signposts and a compact worked example to aid readers. Overall, the work is both correct and highly significant.