{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T11:23:12Z","timestamp":1761218592420,"version":"3.37.3"},"reference-count":14,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,7,1]],"date-time":"2023-07-01T00:00:00Z","timestamp":1688169600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,7,1]],"date-time":"2023-07-01T00:00:00Z","timestamp":1688169600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["1764034"],"award-info":[{"award-number":["1764034"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2024,6]]},"abstract":"Abstract<\/jats:title>A well-known open problem of Meir and Moser asks if the squares of sidelength 1\/n<\/jats:italic> for $$n\\ge 2$$<\/jats:tex-math>\n \n n<\/mml:mi>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be packed perfectly into a rectangle of area $$\\sum _{n=2}^\\infty n^{-2}=\\pi ^2\/6-1$$<\/jats:tex-math>\n \n \n \u2211<\/mml:mo>\n \n n<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n \u221e<\/mml:mi>\n <\/mml:msubsup>\n \n n<\/mml:mi>\n \n -<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n =<\/mml:mo>\n \n \u03c0<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \/<\/mml:mo>\n 6<\/mml:mn>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper we show that for any $$1\/2<t<1$$<\/jats:tex-math>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <<\/mml:mo>\n t<\/mml:mi>\n <<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and any $$n_0$$<\/jats:tex-math>\n \n n<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that is sufficiently large depending on\u00a0t<\/jats:italic>, the squares of sidelength $$n^{-t}$$<\/jats:tex-math>\n \n n<\/mml:mi>\n \n -<\/mml:mo>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for $$n\\ge n_0$$<\/jats:tex-math>\n \n n<\/mml:mi>\n \u2265<\/mml:mo>\n \n n<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be packed perfectly into a square of area $$\\sum _{n=n_0}^\\infty n^{-2t}$$<\/jats:tex-math>\n \n \n \u2211<\/mml:mo>\n \n n<\/mml:mi>\n =<\/mml:mo>\n \n n<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n \u221e<\/mml:mi>\n <\/mml:msubsup>\n \n n<\/mml:mi>\n \n -<\/mml:mo>\n 2<\/mml:mn>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This was previously known (if one packs a rectangle instead of a square) for $$1\/2<t\\le 2\/3$$<\/jats:tex-math>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <<\/mml:mo>\n t<\/mml:mi>\n \u2264<\/mml:mo>\n 2<\/mml:mn>\n \/<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (in which case one can take $$n_0=1$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>).<\/jats:p>","DOI":"10.1007\/s00454-023-00523-y","type":"journal-article","created":{"date-parts":[[2023,7,1]],"date-time":"2023-07-01T21:01:33Z","timestamp":1688245293000},"page":"1178-1189","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength"],"prefix":"10.1007","volume":"71","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0140-7641","authenticated-orcid":false,"given":"Terence","family":"Tao","sequence":"first","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,1]]},"reference":[{"key":"523_CR1","volume-title":"Research Problems in Discrete Geometry","author":"P Brass","year":"2005","unstructured":"Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. 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