The Coven–Meyerowitz tiling conditions for 3 odd prime factors
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Abstract It is well known that if a finite set $$A\subset \mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization $$A\oplus B=\mathbb {Z}_M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊕</mml:mo> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:mrow> </mml:math> of a finite cyclic group. We are interested in characterizing all finite sets $$A\subset \mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> that have this property. Coven and Meyerowitz (J Algebra 212:161–174, 1999) proposed conditions (T1), (T2) that are sufficient for A to tile, and necessary when the cardinality of A has at most two distinct prime factors. They also proved that (T1) holds for all finite tiles, regardless of size. It is not known whether (T2) must hold for all tilings with no restrictions on the number of prime factors of | A |. We prove that the Coven–Meyerowitz tiling condition (T2) holds for all integer tilings of period $$M=(p_ip_jp_k)^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , where $$p_i,p_j,p_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> are distinct odd primes. The proof also provides a classification of all such tilings.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it