{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T01:47:17Z","timestamp":1781228837455,"version":"3.54.1"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"297","license":[{"start":{"date-parts":[[2016,6,9]],"date-time":"2016-06-09T00:00:00Z","timestamp":1465430400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n For a real number\n \n \n \n \n \n \u03b2\n \n <\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \\beta >1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , Erd\u0151s, Jo\u00f3 and Komornik study distances between consecutive points in the set\n \n \\[\n \n \n \n \n X<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b2\n \n <\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n {<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n \n \u2211\n \n <\/mml:mo>\n \n j<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:munderover>\n \n a<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n \n \n \u03b2\n \n <\/mml:mi>\n j<\/mml:mi>\n <\/mml:msup>\n :<\/mml:mo>\n n<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n N<\/mml:mi>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n \n a<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n \n \u2208\n \n <\/mml:mo>\n {<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n m<\/mml:mi>\n }<\/mml:mo>\n \n \n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n .<\/mml:mo>\n <\/mml:mrow>\n X^m(\\beta )=\\Bigl \\{\\sum _{j=0}^n a_j \\beta ^j : n\\in \\mathbb {N},\\,a_j\\in \\{0,1,\\dots ,m\\}\\Bigr \\}.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n Pisot numbers play a crucial role for the properties of\n \n \n \n \n \n X<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b2\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n X^m(\\beta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Following the work of Za\u00efmi, who considered\n \n \n \n \n \n X<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n X^m(\\gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n C<\/mml:mi>\n <\/mml:mrow>\n \n \u2216\n \n <\/mml:mo>\n \n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \\gamma \\in \\mathbb {C}\\setminus \\mathbb {R}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u03b3\n \n <\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n |\\gamma |>1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we show that for any non-real\n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \\gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n m<\/mml:mi>\n ><\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u03b3\n \n <\/mml:mi>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n m>|\\gamma |^2-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the set\n \n \n \n \n \n X<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n X^m(\\gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is not relatively dense in the complex plane.\n <\/p>\n

\n Then we focus on complex Pisot units\n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \\gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with a positive real conjugate\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n \\gamma \u2019<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n m<\/mml:mi>\n ><\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u03b3\n \n <\/mml:mi>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n m>|\\gamma |^2-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . If the number\n \n \n \n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \n \u03b3\n \n <\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n 1\/\\gamma \u2019<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n satisfies Property\u00a0(F), we deduce that\n \n \n \n \n \n X<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n X^m(\\gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is uniformly discrete and relatively dense, i.e.,\n \n \n \n \n \n X<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n X^m(\\gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a Delone set. Moreover, we present an algorithm for determining two parameters of the Delone set\n \n \n \n \n \n X<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n X^m(\\gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which are analogous to minimal and maximal distances in the real case\n \n \n \n \n \n X<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b2\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n X^m(\\beta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For\n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \\gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n satisfying\n \n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\gamma ^3+\\gamma ^2+\\gamma -1=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , explicit formulas for the two parameters are given.\n <\/p>","DOI":"10.1090\/mcom\/2983","type":"journal-article","created":{"date-parts":[[2015,6,9]],"date-time":"2015-06-09T15:23:49Z","timestamp":1433863429000},"page":"401-421","source":"Crossref","is-referenced-by-count":8,"title":["Spectral properties of cubic complex Pisot units"],"prefix":"10.1090","volume":"85","author":[{"given":"Tom\u00e1\u0161","family":"Hejda","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Edita","family":"Pelantov\u00e1","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2015,6,9]]},"reference":[{"key":"1","isbn-type":"print","first-page":"11","article-title":"Cubic Pisot units with finite beta expansions","author":"Akiyama, Shigeki","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3110163047"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/j.jnt.2004.02.001","article-title":"A certain finiteness property of Pisot number systems","volume":"107","author":"Akiyama, Shigeki","year":"2004","journal-title":"J. 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