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Packing Tails of Reciprocal Rectangles into Squares of Equal Area — arXiv:2609.28791

Source

arXiv (en)

Original

https://arxiv.org/abs/2609.28791

License (as archived)

CC0-1.0

Archived

2026-10-08T06:41:09.780Z

SHA-256 of the archived text

e6f4344c6ed95adc652648364ec15f6a23fd8f0d17d47b204d2a9296d6160c47

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How to cite

Packing Tails of Reciprocal Rectangles into Squares of Equal Area — arXiv:2609.28791 — arXiv (en), https://arxiv.org/abs/2609.28791 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bb2fe2ef1acb (Metraipolis archive dc_bb2fe2ef1acb, sha256 e6f4344c6ed95adc652648364ec15f6a23fd8f0d17d47b204d2a9296d6160c47)

@misc{dcbb2fe2ef1acb,
  title = {Packing Tails of Reciprocal Rectangles into Squares of Equal Area — arXiv:2609.28791},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.28791},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bb2fe2ef1acb; sha256 e6f4344c6ed95adc652648364ec15f6a23fd8f0d17d47b204d2a9296d6160c47},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.28791. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The Meir--Moser rectangle-packing problem asks whether all rectangles with side lengths \(1/n\) and \(1/(n+1)\), for \(n\ge1\), can be packed into the unit square with pairwise disjoint interiors. We establish a tail version of this problem. Let \(R_n\) denote the rectangle with these side lengths. We prove that there exists an integer \(m_0\) such that, for every \(m\ge m_0\), the family \(\{R_n:n\ge m\}\) admits a packing, by translations and right-angle rotations, into a square of side length \(m^{-1/2}\), with pairwise disjoint interiors. The area of the square equals the sum of the areas of all the rectangles. The geometric construction recursively decomposes rectangular gaps, while local randomized quotas and random permutations assign subsequent integer indices. We separately control the total area of waiting gaps and the assignment load at each index. The proof is organized in six steps: a finite-prefix reduction, geometric row decompositions, an area bootstrap, a sharp source-load estimate, control of the actual adaptive construction, and a compactness limit. For every finite time horizon, the probability of failure has a bound that is independent of the horizon and can be made arbitrarily small. The adaptive step uses a permanent load ledger, actual fresh height queries, and a one-sided comparison with a frozen source experiment. Compactness then yields an infinite packing. The final packing statements have been checked in Lean 4. A sufficient threshold is \(m_0=10^{1000}\). This result applies only to sufficiently late tails and does not resolve the original Meir--Moser rectangle-packing problem for the full sequence starting at \(n=1\), which remains open.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-25 in math.CO).
Categorie arXiv: Discrete Mathematics (cs.DM) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.28791 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T06:02:10.117Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-25?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).