{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T04:51:53Z","timestamp":1774500713291,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T00:00:00Z","timestamp":1539648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2017078374"],"award-info":[{"award-number":["2017078374"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["17K05159, 17H02849, BBD30028"],"award-info":[{"award-number":["17K05159, 17H02849, BBD30028"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Japan Society for the Promotion of Science and Austrian Science Fund","award":["13346"],"award-info":[{"award-number":["13346"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider Pisot family substitution tilings in      R  d     whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space      R  m     for some integer     m \u2208 N     defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in    R   , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.<\/jats:p>","DOI":"10.3390\/sym10100511","type":"journal-article","created":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T11:07:51Z","timestamp":1539688071000},"page":"511","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Cut-and-Project Schemes for Pisot Family Substitution Tilings"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3587-0992","authenticated-orcid":false,"given":"Jeong-Yup","family":"Lee","sequence":"first","affiliation":[{"name":"Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701, Korea, or KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7549-6686","authenticated-orcid":false,"given":"Shigeki","family":"Akiyama","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan"}]},{"given":"Yasushi","family":"Nagai","sequence":"additional","affiliation":[{"name":"Lehrstuhl f\u00fcr Mathematik und Statistik, Department Mathematik und Informationstechnologie, Montanuniversit\u00e4t, Franz Josef Strasse 18, A-8700 Leoben, Austria"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/978-3-0348-0903-0_2","article-title":"On the Pisot substitution conjecture","volume":"309","author":"Akiyama","year":"2015","journal-title":"Prog. 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