{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:02:39Z","timestamp":1776830559788,"version":"3.51.2"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Given a 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 , the\n beta-transformation<\/italic>\n \n \n \n \n T<\/mml:mi>\n =<\/mml:mo>\n \n T<\/mml:mi>\n \n \n \u03b2\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n T =T_{\\beta }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is defined for\n \n \n \n \n x<\/mml:mi>\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 <\/mml:mrow>\n x \\in [0,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by\n \n \n \n \n T<\/mml:mi>\n x<\/mml:mi>\n :=<\/mml:mo>\n \n \u03b2\n \n <\/mml:mi>\n x<\/mml:mi>\n <\/mml:mrow>\n Tx := \\beta x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (mod 1). The number\n \n \n \n \n \u03b2\n \n <\/mml:mi>\n \\beta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is said to be a beta-number if the orbit\n \n \n \n \n {<\/mml:mo>\n \n T<\/mml:mi>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n }<\/mml:mo>\n <\/mml:mrow>\n \\{T^{n}(1)\\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is finite, hence eventually periodic. In this case\n \n \n \n \n \u03b2\n \n <\/mml:mi>\n \\beta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the root of a monic polynomial\n \n \n \n \n R<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with integer coefficients called the characteristic polynomial of\n \n \n \n \n \u03b2\n \n <\/mml:mi>\n \\beta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . If\n \n \n \n \n P<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n P(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the minimal polynomial of\n \n \n \n \n \u03b2\n \n <\/mml:mi>\n \\beta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then\n \n \n \n \n R<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n P<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n Q<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R(x) = P(x)Q(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for some polynomial\n \n \n \n \n Q<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is the factor\n \n \n \n \n Q<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which concerns us here in case\n \n \n \n \n \u03b2\n \n <\/mml:mi>\n \\beta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether\n \n \n \n \n Q<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n must be cyclotomic in this case, particularly if\n \n \n \n \n 1<\/mml:mn>\n ><\/mml:mo>\n \n \u03b2\n \n <\/mml:mi>\n ><\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n 1 > \\beta > 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in\n \n \n \n \n [<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1.9324<\/mml:mn>\n ]<\/mml:mo>\n \n \u222a\n \n <\/mml:mo>\n [<\/mml:mo>\n 1.9333<\/mml:mn>\n ,<\/mml:mo>\n 1.96<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [1,1.9324]\\cup [1.9333,1.96]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (an infinite set), by a search up to degree\n \n \n \n 50<\/mml:mn>\n 50<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n [<\/mml:mo>\n 1.9<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [1.9,2]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , to degree\n \n \n \n 60<\/mml:mn>\n 60<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n [<\/mml:mo>\n 1.96<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [1.96,2]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and to degree\n \n \n \n 20<\/mml:mn>\n 20<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n [<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n 2.2<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [2,2.2]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We find the smallest counterexample, the counterexample of smallest degree, examples where\n \n \n \n \n Q<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is nonreciprocal, and examples where\n \n \n \n \n Q<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n from above, and infinite sequences of\n \n \n \n \n \u03b2\n \n <\/mml:mi>\n \\beta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n Q<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n nonreciprocal which converge to\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n from below and to the\n \n \n \n 6<\/mml:mn>\n 6<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in\n \n \n \n \n [<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [1,2]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The Pisot numbers for which\n \n \n \n \n Q<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n Q(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is cyclotomic are related to an interesting closed set of numbers\n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {F}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n introduced by Flatto, Lagarias and Poonen in connection with the zeta function of\n \n \n \n T<\/mml:mi>\n T<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Our examples show that the set\n \n \n \n S<\/mml:mi>\n S<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of Pisot numbers is not a subset of\n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {F}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-96-00693-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"841-860","source":"Crossref","is-referenced-by-count":18,"title":["On beta expansions for Pisot numbers"],"prefix":"10.1090","volume":"65","author":[{"given":"David","family":"Boyd","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","doi-asserted-by":"crossref","first-page":"215","DOI":"10.24033\/asens.1156","article-title":"Ensembles ferm\u00e9s de nombres alg\u00e9briques","volume":"83","author":"Amara, Mohamed","year":"1966","journal-title":"Ann. 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