{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T10:02:35Z","timestamp":1751536955720,"version":"3.37.3"},"reference-count":14,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,7,16]],"date-time":"2022-07-16T00:00:00Z","timestamp":1657929600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,7,16]],"date-time":"2022-07-16T00:00:00Z","timestamp":1657929600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100009232","name":"University of Debrecen","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100009232","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$A_{\\varphi }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mi>\u03c6<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denote the matrix of rotation with angle <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varphi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c6<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of the Euclidean plane, FLOOR the function which rounds a real point to the nearest lattice point down on the left and ROUND the function for rounding off a vector to the nearest node of the lattice. We prove under the natural assumption <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varphi \\not = k\\frac{\\pi }{2}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03c6<\/mml:mi>\n                    <mml:mo>\u2260<\/mml:mo>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mfrac>\n                      <mml:mi>\u03c0<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mfrac>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that the functions <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\mathrm{FLOOR}\\,}}\\circ A_{\\varphi }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mi>FLOOR<\/mml:mi>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>\u2218<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>\u03c6<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\mathrm{ROUND}\\,}}\\circ A_{\\varphi }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mi>ROUND<\/mml:mi>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>\u2218<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>\u03c6<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are neither surjective nor injective. More precisely we prove lower and upper estimates for the size of the sets of lattice points, which are the image of two lattice points as well as of lattice points, which have no preimages. It turns out that the densities of those sets are positive.<\/jats:p>","DOI":"10.1007\/s10998-022-00480-8","type":"journal-article","created":{"date-parts":[[2022,7,16]],"date-time":"2022-07-16T12:02:43Z","timestamp":1657972963000},"page":"564-577","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Rotation on the digital plane"],"prefix":"10.1007","volume":"86","author":[{"given":"Carolin","family":"Hannusch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Attila","family":"Peth\u0151","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,7,16]]},"reference":[{"issue":"3","key":"480_CR1","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/s10474-005-0221-z","volume":"108","author":"S Akiyama","year":"2005","unstructured":"S. Akiyama, T. Borb\u00e9ly, H. Brunotte, A. Peth\u0151, J.M. Thuswaldner, Generalized radix representations and dynamical systems I. Acta Math. Hung. 108(3), 207\u2013238 (2005)","journal-title":"Acta Math. Hung."},{"key":"480_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10998-006-0002-7","volume":"52","author":"S Akiyama","year":"2006","unstructured":"S. Akiyama, H. Brunotte, A. Peth\u0151, W. Steiner, Remarks on a conjecture on certain integer sequences. Period. Math. Hung. 52, 1\u201317 (2006)","journal-title":"Period. Math. Hung."},{"issue":"1","key":"480_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.21099\/tkbjm\/1496165198","volume":"32","author":"S Akiyama","year":"2008","unstructured":"S. Akiyama, H. Brunotte, A. Peth\u0151, W. Steiner, Periodicity of certain piecewise affine planar maps. Tsukuba J. Math. 32(1), 1\u201355 (2008)","journal-title":"Tsukuba J. 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