{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:46:32Z","timestamp":1740109592189,"version":"3.37.3"},"reference-count":9,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T00:00:00Z","timestamp":1689033600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T00:00:00Z","timestamp":1689033600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100009567","name":"Budapest University of Technology and Economics","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100009567","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2024,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A sublattice of the three-dimensional integer lattice <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {Z}}^3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>Z<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is called cubic sublattice if there exists a basis of the sublattice whose elements are pairwise orthogonal and of equal lengths. We show that for an integer vector <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\textbf{v}}\\in {\\mathbb {Z}}^3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>v<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>Z<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> whose squared length is divisible by\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$d^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, there exists a cubic sublattice containing <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\textbf{v}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>v<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with edge length\u00a0<jats:italic>d<\/jats:italic>. This improves one of the main result of a paper of Goswick et al. (J. Number Theory <jats:bold>132<\/jats:bold>(1), 37\u201353 (2012)), where similar theorem was proved by using the decomposition theory of Hurwitz integral quaternions. We give an elementary proof heavily using cross product. This method allows us to characterize the cubic sublattices.<\/jats:p>","DOI":"10.1007\/s00454-023-00535-8","type":"journal-article","created":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T14:02:03Z","timestamp":1689084123000},"page":"1369-1380","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Cubic Sublattices"],"prefix":"10.1007","volume":"71","author":[{"given":"M\u00e1rton","family":"Horv\u00e1th","sequence":"first","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,11]]},"reference":[{"key":"535_CR1","volume-title":"Diophantine Analysis","author":"RD Carmichael","year":"1915","unstructured":"Carmichael, R.D.: Diophantine Analysis. Wiley, New York (1915)"},{"key":"535_CR2","volume-title":"An Introduction to the Geometry of Numbers. 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Freeman, New York (1985)"},{"issue":"4","key":"535_CR6","first-page":"525","volume":"49","author":"EW Kiss","year":"2012","unstructured":"Kiss, E.W., Kutas, P.: Cubes of integral vectors in dimension four. Studia Sci. Math. Hung. 49(4), 525\u2013537 (2012)","journal-title":"Studia Sci. Math. Hung."},{"issue":"2","key":"535_CR7","doi-asserted-by":"publisher","first-page":"118","DOI":"10.4169\/college.math.j.42.2.118","volume":"42","author":"R Parris","year":"2011","unstructured":"Parris, R.: Lattice cubes. College Math. J. 42(2), 118\u2013125 (2011)","journal-title":"College Math. J."},{"key":"535_CR8","unstructured":"S\u00e1rk\u00f6zy, A.: Lattice cubes in $$3$$-space. Mat. Lapok 12, 232\u2013245 (1961). (in Hungarian)"},{"issue":"5","key":"535_CR9","first-page":"360","volume":"69","author":"R Spira","year":"1962","unstructured":"Spira, R.: The Diophantine equation $$x^{2}+y^{2}+z^{2}=m^{2}$$. Am. Math. Mon. 69(5), 360\u2013365 (1962)","journal-title":"Am. Math. 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