Pub Date : 2022-05-20DOI: 10.1017/S1446788722000088
Sebastián Tapia-García
Abstract Savin [‘ $mathcal {C}^{1}$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal. 3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations 274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.
{"title":"REGULARITY OF AML FUNCTIONS IN TWO-DIMENSIONAL NORMED SPACES","authors":"Sebastián Tapia-García","doi":"10.1017/S1446788722000088","DOIUrl":"https://doi.org/10.1017/S1446788722000088","url":null,"abstract":"Abstract Savin [‘ \u0000$mathcal {C}^{1}$\u0000 regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal. 3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations 274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"1 1","pages":"406 - 430"},"PeriodicalIF":0.7,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78210187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-11DOI: 10.1017/S1446788722000155
P. Ara
Abstract In this paper, we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra $L(E,omega )$ of a row-finite vertex weighted graph $(E,omega )$ is $*$ -isomorphic to the lower Leavitt path algebra of a certain bipartite separated graph $(E(omega ),C(omega ))$ . For a general locally finite weighted graph $(E, omega )$ , we show that a certain quotient $L_1(E,omega )$ of $L(E,omega )$ is $*$ -isomorphic to an upper Leavitt path algebra of another bipartite separated graph $(E(w)_1,C(w)^1)$ . We furthermore introduce the algebra ${L^{mathrm {ab}}} (E,w)$ , which is a universal tame $*$ -algebra generated by a set of partial isometries. We draw some consequences of our results for the structure of ideals of $L(E,omega )$ , and we study in detail two different maximal ideals of the Leavitt algebra $L(m,n)$ .
{"title":"LEAVITT PATH ALGEBRAS OF WEIGHTED AND SEPARATED GRAPHS","authors":"P. Ara","doi":"10.1017/S1446788722000155","DOIUrl":"https://doi.org/10.1017/S1446788722000155","url":null,"abstract":"Abstract In this paper, we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra \u0000$L(E,omega )$\u0000 of a row-finite vertex weighted graph \u0000$(E,omega )$\u0000 is \u0000$*$\u0000 -isomorphic to the lower Leavitt path algebra of a certain bipartite separated graph \u0000$(E(omega ),C(omega ))$\u0000 . For a general locally finite weighted graph \u0000$(E, omega )$\u0000 , we show that a certain quotient \u0000$L_1(E,omega )$\u0000 of \u0000$L(E,omega )$\u0000 is \u0000$*$\u0000 -isomorphic to an upper Leavitt path algebra of another bipartite separated graph \u0000$(E(w)_1,C(w)^1)$\u0000 . We furthermore introduce the algebra \u0000${L^{mathrm {ab}}} (E,w)$\u0000 , which is a universal tame \u0000$*$\u0000 -algebra generated by a set of partial isometries. We draw some consequences of our results for the structure of ideals of \u0000$L(E,omega )$\u0000 , and we study in detail two different maximal ideals of the Leavitt algebra \u0000$L(m,n)$\u0000 .","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"191 1","pages":"1 - 25"},"PeriodicalIF":0.7,"publicationDate":"2022-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85186930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-10DOI: 10.1017/s144678872200009x
Apurva Seth, P. Vaidyanathan
We show that the properties of being rationally K-stable passes from the fibres of a continuous $C(X)$ -algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers.
{"title":"RATIONAL -STABILITY OF CONTINUOUS -ALGEBRAS","authors":"Apurva Seth, P. Vaidyanathan","doi":"10.1017/s144678872200009x","DOIUrl":"https://doi.org/10.1017/s144678872200009x","url":null,"abstract":"\u0000 We show that the properties of being rationally K-stable passes from the fibres of a continuous \u0000 \u0000 \u0000 \u0000$C(X)$\u0000\u0000 \u0000 -algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"39 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78369372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-10DOI: 10.1017/s144678872100029x
{"title":"JAZ volume 112 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s144678872100029x","DOIUrl":"https://doi.org/10.1017/s144678872100029x","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"12 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83440036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-10DOI: 10.1017/s1446788721000306
{"title":"INDEX","authors":"","doi":"10.1017/s1446788721000306","DOIUrl":"https://doi.org/10.1017/s1446788721000306","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"33 1","pages":"431 - 432"},"PeriodicalIF":0.7,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80100733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-10DOI: 10.1017/s1446788721000288
{"title":"JAZ volume 112 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s1446788721000288","DOIUrl":"https://doi.org/10.1017/s1446788721000288","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"61 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80548847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-13DOI: 10.1017/s1446788723000010
Carl-Fredrik Nyberg Brodda
This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman [‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc.136(3) (2004), 513–524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a $mathbf {C}$ -multiplication table, where $mathbf {C}$ is any reversal-closed super- $operatorname {mathrm {AFL}}$ . In particular, we deduce that the free product of two groups with $mathbf {ET0L}$ with respect to indexed multiplication tables again has an $mathbf {ET0L}$ with respect to an indexed multiplication table.
{"title":"MULTIPLICATION TABLES AND WORD-HYPERBOLICITY IN FREE PRODUCTS OF SEMIGROUPS, MONOIDS AND GROUPS","authors":"Carl-Fredrik Nyberg Brodda","doi":"10.1017/s1446788723000010","DOIUrl":"https://doi.org/10.1017/s1446788723000010","url":null,"abstract":"\u0000 This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman [‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc.136(3) (2004), 513–524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a \u0000 \u0000 \u0000 \u0000$mathbf {C}$\u0000\u0000 \u0000 -multiplication table, where \u0000 \u0000 \u0000 \u0000$mathbf {C}$\u0000\u0000 \u0000 is any reversal-closed super-\u0000 \u0000 \u0000 \u0000$operatorname {mathrm {AFL}}$\u0000\u0000 \u0000 . In particular, we deduce that the free product of two groups with \u0000 \u0000 \u0000 \u0000$mathbf {ET0L}$\u0000\u0000 \u0000 with respect to indexed multiplication tables again has an \u0000 \u0000 \u0000 \u0000$mathbf {ET0L}$\u0000\u0000 \u0000 with respect to an indexed multiplication table.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"69 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89099491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-06DOI: 10.1017/S1446788722000027
R. Grigorchuk, D. Savchuk
Abstract The ring $mathbb Z_{d}$ of d-adic integers has a natural interpretation as the boundary of a rooted d-ary tree $T_{d}$ . Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from $mathbb Z_{d}$ to itself. In the case when $d=p$ is prime, Anashin [‘Automata finiteness criterion in terms of van der Put series of automata functions’,p-Adic Numbers Ultrametric Anal. Appl. 4(2) (2012), 151–160] showed that $fin mathrm {Lip}^{1}(mathbb Z_{p})$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a p-automatic sequence over a finite subset of $mathbb Z_{p}cap mathbb Q$ . We generalize this result to arbitrary integers $dgeq 2$ and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other and vice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.
d进整数的环$mathbb Z_{d}$可以很自然地解释为有根的d进树$T_{d}$的边界。该树的自同态(即螺线线映射)与从$mathbb Z_{d}$到自身的1-Lipschitz映射是一一对应的。在$d=p$为素数的情况下，Anashin[关于自动机函数的van der Put级数的自动机有限性判据]，p进数超度量。应用程序4(2)(2012)，151-160]表明$fin mathrm {Lip}^{1}(mathbb Z_{p})$是由有限Mealy自动机定义的，当且仅当其van der Put级数的约简系数构成$mathbb Z_{p}cap mathbb Q$有限子集上的p-自动序列。我们将这一结果推广到任意整数$dgeq 2$，并描述了产生这样一个序列的摩尔自动机与产生相应根树自同态的米利自动机之间的显式联系。我们还生成了两种将一个自动机转换为另一个自动机的算法，反之亦然。作为演示，我们将我们的算法应用于Thue-Morse序列和作用于二叉根树的lamplighter群的一个生成器。
{"title":"SOLENOIDAL MAPS, AUTOMATIC SEQUENCES, VAN DER PUT SERIES, AND MEALY AUTOMATA","authors":"R. Grigorchuk, D. Savchuk","doi":"10.1017/S1446788722000027","DOIUrl":"https://doi.org/10.1017/S1446788722000027","url":null,"abstract":"Abstract The ring $mathbb Z_{d}$ of d-adic integers has a natural interpretation as the boundary of a rooted d-ary tree $T_{d}$ . Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from $mathbb Z_{d}$ to itself. In the case when $d=p$ is prime, Anashin [‘Automata finiteness criterion in terms of van der Put series of automata functions’,p-Adic Numbers Ultrametric Anal. Appl. 4(2) (2012), 151–160] showed that $fin mathrm {Lip}^{1}(mathbb Z_{p})$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a p-automatic sequence over a finite subset of $mathbb Z_{p}cap mathbb Q$ . We generalize this result to arbitrary integers $dgeq 2$ and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other and vice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"55 1","pages":"78 - 109"},"PeriodicalIF":0.7,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74849763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-22DOI: 10.1017/S1446788722000039
Ching-on Lo, A. Loh
Abstract Let u and $varphi $ be two analytic functions on the unit disk D such that $varphi (D) subset D$ . A weighted composition operator $uC_{varphi }$ induced by u and $varphi $ is defined on $A^2_{alpha }$ , the weighted Bergman space of D, by $uC_{varphi }f := u cdot f circ varphi $ for every $f in A^2_{alpha }$ . We obtain sufficient conditions for the compactness of $uC_{varphi }$ in terms of function-theoretic properties of u and $varphi $ . We also characterize when $uC_{varphi }$ on $A^2_{alpha }$ is Hilbert–Schmidt. In particular, the characterization is independent of $alpha $ when $varphi $ is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on $A^2_{alpha }$ .
设u和$varphi $为单位圆盘D上的两个解析函数，使得$varphi (D) subset D$。对于每一个$f in A^2_{alpha }$，在D的加权Bergman空间$A^2_{alpha }$上，通过$uC_{varphi }f := u cdot f circ varphi $定义由u和$varphi $诱导的加权复合算子$uC_{varphi }$。利用u和$varphi $的泛函性质，得到了$uC_{varphi }$紧性的充分条件。我们还描述了$A^2_{alpha }$上的$uC_{varphi }$是Hilbert-Schmidt。特别地，当$varphi $是d的自同构时，表征与$alpha $无关。进一步，我们研究了$A^2_{alpha }$上两个加权复合算子的Hilbert-Schmidt差分。
{"title":"COMPACT AND HILBERT–SCHMIDT WEIGHTED COMPOSITION OPERATORS ON WEIGHTED BERGMAN SPACES","authors":"Ching-on Lo, A. Loh","doi":"10.1017/S1446788722000039","DOIUrl":"https://doi.org/10.1017/S1446788722000039","url":null,"abstract":"Abstract Let u and \u0000$varphi $\u0000 be two analytic functions on the unit disk D such that \u0000$varphi (D) subset D$\u0000 . A weighted composition operator \u0000$uC_{varphi }$\u0000 induced by u and \u0000$varphi $\u0000 is defined on \u0000$A^2_{alpha }$\u0000 , the weighted Bergman space of D, by \u0000$uC_{varphi }f := u cdot f circ varphi $\u0000 for every \u0000$f in A^2_{alpha }$\u0000 . We obtain sufficient conditions for the compactness of \u0000$uC_{varphi }$\u0000 in terms of function-theoretic properties of u and \u0000$varphi $\u0000 . We also characterize when \u0000$uC_{varphi }$\u0000 on \u0000$A^2_{alpha }$\u0000 is Hilbert–Schmidt. In particular, the characterization is independent of \u0000$alpha $\u0000 when \u0000$varphi $\u0000 is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on \u0000$A^2_{alpha }$\u0000 .","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"404 1","pages":"208 - 225"},"PeriodicalIF":0.7,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76482642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-21DOI: 10.1017/s1446788721000422
J. BECK, W. W. L. CHEN
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 $mathcal {P}$ .
{"title":"RETRACTED - THE KRONECKER–WEYL EQUIDISTRIBUTION THEOREM AND GEODESICS IN 3-MANIFOLDS","authors":"J. BECK, W. W. L. CHEN","doi":"10.1017/s1446788721000422","DOIUrl":"https://doi.org/10.1017/s1446788721000422","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>\u0000$3$\u0000</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>\u0000$mathcal {P}$\u0000</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>\u0000$mathcal {P}$\u0000</span></span></span></span>.</p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"20 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$3$ -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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