{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:49:23Z","timestamp":1776804563513,"version":"3.51.2"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"315","license":[{"start":{"date-parts":[[2019,4,5]],"date-time":"2019-04-05T00:00:00Z","timestamp":1554422400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper we address the task of finding well approximating lattices for a given finite set\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of points in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R Superscript n\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathbb R}^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    motivated by practical texture analytic problems. More precisely, we search for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold-italic o comma bold-italic d bold 1 comma ellipsis comma bold-italic d Subscript bold-italic n Baseline element-of double-struck upper R Superscript n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">o<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"bold-italic\">d<\/mml:mi>\n                              <mml:mn mathvariant=\"bold\">1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"bold-italic\">d<\/mml:mi>\n                              <mml:mi mathvariant=\"bold-italic\">n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\boldsymbol {o},\\boldsymbol {d_1}, \\dots ,\\boldsymbol {d_n}\\in \\mathbb {R}^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold-italic a minus bold-italic o\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">a<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"bold-italic\">o<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\boldsymbol {a}-\\boldsymbol {o}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is close to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Lamda equals bold-italic d bold 1 double-struck upper Z plus midline-horizontal-ellipsis plus bold-italic d Subscript bold-italic n Baseline double-struck upper Z\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u039b\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"bold-italic\">d<\/mml:mi>\n                              <mml:mn mathvariant=\"bold\">1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"bold-italic\">d<\/mml:mi>\n                              <mml:mi mathvariant=\"bold-italic\">n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Lambda =\\boldsymbol {d_1}\\mathbb {Z}+\\dots +\\boldsymbol {d_n}\\mathbb {Z}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for every\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold-italic a element-of upper A\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">a<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mi>A<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\boldsymbol {a}\\in A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . First we deal with the one-dimensional case, where we show that in a sense the results are almost the best possible. These results easily extend to the multi-dimensional case where the directions of the axes are given, too. Thereafter we treat the general multi-dimensional case. Our method relies on the LLL algorithm. Finally, we apply the least squares algorithm to optimize the results. We give several examples to illustrate our approach.\n                  <\/p>","DOI":"10.1090\/mcom\/3320","type":"journal-article","created":{"date-parts":[[2017,8,30]],"date-time":"2017-08-30T10:04:26Z","timestamp":1504087466000},"page":"369-387","source":"Crossref","is-referenced-by-count":2,"title":["Finding well approximating lattices for a finite set of points"],"prefix":"10.1090","volume":"88","author":[{"given":"A.","family":"Hajdu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L.","family":"Hajdu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"Tijdeman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2018,4,5]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"167","DOI":"10.2140\/obs.2013.1.167","article-title":"Finding simultaneous Diophantine approximations with prescribed quality","author":"Bosma, Wieb","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781935107019"},{"key":"2","series-title":"Cambridge Tracts in Mathematics and Mathematical Physics, No. 45","volume-title":"An introduction to Diophantine approximation","author":"Cassels, J. W. S.","year":"1957"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"271","DOI":"10.2140\/obs.2013.1.271","article-title":"Approximate common divisors via lattices","author":"Cohn, Henry","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781935107019"},{"key":"4","doi-asserted-by":"crossref","unstructured":"R. M. Haralick, K. Shanmugam and I. Dinstein, Textural features for image classification, IEEE Trans. on Systems, Man, and Cybern., 6 (1973), 610\u2013621.","DOI":"10.1109\/TSMC.1973.4309314"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/3-540-44670-2_6","article-title":"Approximate integer common divisors","author":"Howgrave-Graham, Nick","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/3540424881"},{"issue":"4","key":"6","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1007\/BF01457454","article-title":"Factoring polynomials with rational coefficients","volume":"261","author":"Lenstra, A. K.","year":"1982","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"7","series-title":"Lecture Notes in Mathematics","isbn-type":"print","volume-title":"Diophantine approximation","volume":"785","author":"Schmidt, Wolfgang M.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/3540097627"},{"key":"8","unstructured":"L. G. Shapiro and G. C. Stockman, Computer Vision, Prentice Hall, Upper Saddle River, 2001."},{"key":"9","unstructured":"W. Steurer and S. Deloudi, Crystallography of Quasicrystals: Concepts, Methods and Structures, Springer, Heidelberg, 2009."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2019-88-315\/S0025-5718-2018-03320-8\/S0025-5718-2018-03320-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-315\/S0025-5718-2018-03320-8\/S0025-5718-2018-03320-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:54:31Z","timestamp":1776801271000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-315\/S0025-5718-2018-03320-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,5]]},"references-count":9,"journal-issue":{"issue":"315","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["S0025-5718-2018-03320-8"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3320","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2018,4,5]]}}}