{"title":"RETRACTED - THE KRONECKER–WEYL EQUIDISTRIBUTION THEOREM AND GEODESICS IN 3-MANIFOLDS","authors":"J. BECK, W. W. L. CHEN","doi":"10.1017/s1446788721000422","DOIUrl":null,"url":null,"abstract":"<p>Given any rectangular polyhedron <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220617152558528-0118:S1446788721000422:S1446788721000422_inline1.png\"/><span data-mathjax-type=\"texmath\"><span>\n$3$\n</span></span></span></span>-manifold <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220617152558528-0118:S1446788721000422:S1446788721000422_inline2.png\"/><span data-mathjax-type=\"texmath\"><span>\n$\\mathcal {P}$\n</span></span></span></span> tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220617152558528-0118:S1446788721000422:S1446788721000422_inline3.png\"/><span data-mathjax-type=\"texmath\"><span>\n$\\mathcal {P}$\n</span></span></span></span>.</p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"20 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2022-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Australian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s1446788721000422","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Given any rectangular polyhedron
$3$
-manifold
$\mathcal {P}$
tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in
$\mathcal {P}$
.
期刊介绍：
The Journal of the Australian Mathematical Society is the oldest journal of the Society, and is well established in its coverage of all areas of pure mathematics and mathematical statistics. It seeks to publish original high-quality articles of moderate length that will attract wide interest. Papers are carefully reviewed, and those with good introductions explaining the meaning and value of the results are preferred.
Published Bi-monthly
Published for the Australian Mathematical Society
$3$
-manifold 
$\mathcal {P}$
tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in 
$\mathcal {P}$
.
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