Packing Tails of Reciprocal Rectangles into Squares of Equal Area
Abstract
The Meir--Moser rectangle-packing problem asks whether all rectangles with side lengths 1/n and 1/(n+1), for n≥1, can be packed into the unit square with pairwise disjoint interiors. We establish a tail version of this problem. Let Rn denote the rectangle with these side lengths. We prove that there exists an integer m0 such that, for every m≥ m0, the family {Rn:n≥ 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 m0=101000. 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.
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BibTeX
@misc{jiang2026packing,
title = {{Packing Tails of Reciprocal Rectangles into Squares of Equal Area}},
author = {Yu Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28791},
eprint = {2609.28791},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28791},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Packing Tails of Reciprocal Rectangles into Squares of Equal Area
AU - Yu Jiang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28791
UR - https://arxiv.org/abs/2609.28791
AB - The Meir--Moser rectangle-packing problem asks whether all rectangles with side lengths 1/n and 1/(n+1), for n≥1, 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≥ m_0, the family {R_n:n≥ 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.
ER -