{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:52:13Z","timestamp":1776844333493,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,11,14]],"date-time":"2002-11-14T00:00:00Z","timestamp":1037232000000},"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":"

\n Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erd\u0151s, Jo\u00f3 and Komornik in 1990, is the determination of\n \n \n \n \n l<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n l(q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for Pisot numbers\n \n \n \n q<\/mml:mi>\n q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \\[\n \n \n \n l<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n inf<\/mml:mo>\n (<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n y<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n :<\/mml:mo>\n y<\/mml:mi>\n =<\/mml:mo>\n \n \n \u03f5\n \n <\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n \u03f5\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n \n \u03f5\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msup>\n ,<\/mml:mo>\n \n \n \u03f5\n \n <\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \n \u2208\n \n <\/mml:mo>\n {<\/mml:mo>\n \n \u00b1\n \n <\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 0<\/mml:mn>\n }<\/mml:mo>\n ,<\/mml:mo>\n y<\/mml:mi>\n \n \u2260\n \n <\/mml:mo>\n 0<\/mml:mn>\n )<\/mml:mo>\n .<\/mml:mo>\n <\/mml:mrow>\n l(q) = \\inf (|y|: y = \\epsilon _0 + \\epsilon _1 q^1 + \\cdots + \\epsilon _n q^n, \\epsilon _i \\in \\{\\pm 1, 0\\}, y \\neq 0).<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n Although the quantity\n \n \n \n \n l<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n l(q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is known for some Pisot numbers\n \n \n \n q<\/mml:mi>\n q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , there has been no general method for computing\n \n \n \n \n l<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n l(q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . This paper gives such an algorithm. With this algorithm, some properties of\n \n \n \n \n l<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n l(q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and its generalizations are investigated. A related question concerns the analogy of\n \n \n \n \n l<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n l(q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , denoted\n \n \n \n \n a<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n a(q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where the coefficients are restricted to\n \n \n \n \n \n \u00b1\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \\pm 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ; in particular, for which non-Pisot numbers is\n \n \n \n \n a<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n a(q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n nonzero? This paper finds an infinite class of Salem numbers where\n \n \n \n \n a<\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n \n \u2260\n \n <\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n a(q) \\neq 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-01-01336-9","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"767-780","source":"Crossref","is-referenced-by-count":19,"title":["Some computations on the spectra of Pisot and Salem numbers"],"prefix":"10.1090","volume":"71","author":[{"given":"Peter","family":"Borwein","sequence":"first","affiliation":[]},{"given":"Kevin","family":"Hare","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,11,14]]},"reference":[{"issue":"144","key":"1","doi-asserted-by":"publisher","first-page":"1244","DOI":"10.2307\/2006349","article-title":"Pisot and Salem numbers in intervals of the real line","volume":"32","author":"Boyd, David W.","year":"1978","journal-title":"Math. 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