2022/02/08 by Terence Tao, Tao, Terence · 1 citation
Materials Science · Mathematics · #52C15 #Combinatorics #FOS: Mathematics #Geometry #Harmonic #Mathematical Approximation and Integration #Mathematics #Metric Geometry (math.MG) #Physics #Point processes and geometric inequalities #Quantum mechanics #Quasicrystal Structures and Properties #Rectangle #Square (algebra) #math.MG #msc:52C15
paper · pdf · doi:10.48550/arxiv.2202.03594
published in arXiv (Cornell University) (Cornell University) · 11 pages, 1 figure. Several minor corrections
openalex publication_date 2022/02/08 · arxiv created 2022/03/10 · arxiv updated 2022/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
A well known open problem of Meir and Moser asks if the squares of sidelength 1/n for n ≥ 2 can be packed perfectly into a square of area ∑n=2^∞ (1)/(n2) = (π2)/(6)-1. In this paper we show that for any 1/2 < t < 1, and any n0 that is sufficiently large depending on t, the squares of sidelength n-t for n ≥ n0 can be packed perfectly into a square of area ∑n=n0^∞ \frac1n2t. This was previously known (if one packs a rectangle instead of a square) for 1/2 < t ≤ 2/3 (in which case one can take n0=1).