Bulletin of the London Mathematical Society最新文献

{"title":"Closed \u0000 \u0000 \u0000 G\u0000 2\u0000 \u0000 $G_2$\u0000 -structures with negative Ricci curvature","authors":"Alec Payne","doi":"10.1112/blms.70029","DOIUrl":"https://doi.org/10.1112/blms.70029","url":null,"abstract":"<p>We study existence problems for closed <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$G_2$</annotation>\u0000 </semantics></math>-structures with negative Ricci curvature, and we prove the <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$G_2$</annotation>\u0000 </semantics></math>-Goldberg conjecture for noncompact manifolds. We first show that no closed manifold admits a closed <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$G_2$</annotation>\u0000 </semantics></math>-structure with negative Ricci curvature. In the noncompact setting, we show that no complete manifold admits a closed <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$G_2$</annotation>\u0000 </semantics></math>-structure with Ricci curvature pinched sufficiently close to a negative constant. As a consequence, an Einstein closed <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$G_2$</annotation>\u0000 </semantics></math>-structure on a complete manifold must be torsion-free. In addition, when the Einstein metric is incomplete, we find restrictions on lengths of geodesics.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1270-1284"},"PeriodicalIF":0.8,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lorentzian polynomials and the independence sequences of graphs","authors":"Amire Bendjeddou,&nbsp;Leonard Hardiman","doi":"10.1112/blms.70031","DOIUrl":"https://doi.org/10.1112/blms.70031","url":null,"abstract":"<p>We study the multivariate independence polynomials of graphs and the log-concavity of the coefficients of their univariate restrictions. Let <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mn>4</mn>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$R_{W_4}$</annotation>\u0000 </semantics></math> be the operator defined on simple and undirected graphs which replaces each edge with a caterpillar of size 4. We prove that all graphs in the image of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mn>4</mn>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$R_{W_4}$</annotation>\u0000 </semantics></math> are what we call pre-Lorentzian, that is, their multivariate independence polynomial becomes Lorentzian after appropriate manipulations. In particular, as pre-Lorentzian graphs have log-concave (and therefore unimodal) independence sequences, our result makes progress on a conjecture of Alavi, Malde, Schwenk and Erdős which asks if the independence sequence of trees or forests is unimodal.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1305-1323"},"PeriodicalIF":0.8,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic intersection strength for hyperbolic surfaces","authors":"Manman Jiang,&nbsp;Huiping Pan","doi":"10.1112/blms.70030","DOIUrl":"https://doi.org/10.1112/blms.70030","url":null,"abstract":"<p>We show that the algebraic intersection strength of hyperbolic surfaces of genus <span></span><math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$g$</annotation>\u0000 </semantics></math> has a minimum in the moduli space. We also describe the asymptotic behavior of the algebraic intersection strength in the moduli space.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1285-1304"},"PeriodicalIF":0.8,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distinguishing internally club and approachable on an Infinite interval","authors":"Hannes Jakob,&nbsp;Maxwell Levine","doi":"10.1112/blms.70013","DOIUrl":"https://doi.org/10.1112/blms.70013","url":null,"abstract":"&lt;p&gt;Krueger showed that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;PFA&lt;/mi&gt;\u0000 &lt;annotation&gt;${textsf {PFA}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; implies that for all regular &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Θ&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℵ&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Theta geqslant aleph _2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, there are stationarily many &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Θ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℵ&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$[H(Theta)]^{aleph _1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$n&lt;omega$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Θ&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℵ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Theta geqslant aleph _{n+1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, there is a stationary subset of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Θ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℵ&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$[H(Theta)]^{aleph _n}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; consisting of sets","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1026-1039"},"PeriodicalIF":0.8,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A generalized pseudo-rotation with positive topological entropy","authors":"Erman Çineli","doi":"10.1112/blms.70021","DOIUrl":"https://doi.org/10.1112/blms.70021","url":null,"abstract":"<p>In this note, we give examples of Hamiltonian diffeomorphisms which are on one hand dynamically complicated, for instance, with positive topological entropy, and on the other hand minimal from the perspective of Floer theory. The minimality is in the sense that the barcode of the Floer complex of all iterates of these maps consists of only infinite bars. In particular, the maps have zero barcode entropy.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1140-1149"},"PeriodicalIF":0.8,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Intrinsic Hopf–Lax formula and Hamilton–Jacobi equation","authors":"Daniela Di Donato","doi":"10.1112/blms.70026","DOIUrl":"https://doi.org/10.1112/blms.70026","url":null,"abstract":"<p>The purpose of this article is to analyze the notion of intrinsic Hopf–Lax formula and its connection with the Hamilton–Jacobi-type equation.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1208-1228"},"PeriodicalIF":0.8,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70026","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A comparison of Hochschild homology in algebraic and smooth settings","authors":"David Kazhdan,&nbsp;Maarten Solleveld","doi":"10.1112/blms.70028","DOIUrl":"https://doi.org/10.1112/blms.70028","url":null,"abstract":"&lt;p&gt;Consider a complex affine variety &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;annotation&gt;$tilde{V}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and a real analytic Zariski-dense submanifold &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;annotation&gt;$V$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;annotation&gt;$tilde{V}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We compare modules over the ring &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {O} (tilde{V})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of regular functions on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 &lt;annotation&gt;$tilde{V}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with modules over the ring &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$C^infty (V)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of smooth complex valued functions on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;annotation&gt;$V$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Under a mild condition on the tangent spaces, we prove that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$C^infty (V)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is flat as a module over &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1249-1269"},"PeriodicalIF":0.8,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ind-étale versus formally étale","authors":"Shubhodip Mondal,&nbsp;Alapan Mukhopadhyay","doi":"10.1112/blms.70025","DOIUrl":"https://doi.org/10.1112/blms.70025","url":null,"abstract":"<p>We show that when <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> is a reduced algebra over a characteristic zero field <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> and the module of Kähler differentials <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Ω</mi>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>/</mo>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$Omega _{A/k}=0$</annotation>\u0000 </semantics></math>, then <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> is ind-étale, partially answering a question of Bhatt. As further applications of this result, we deduce a rigidity property of Hochschild homology and special instances of Weibel's conjecture and Vorst's conjecture without any noetherian assumptions.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1195-1207"},"PeriodicalIF":0.8,"publicationDate":"2025-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Algebraic Kirchberg–Phillips Question for Leavitt path algebras","authors":"Efren Ruiz","doi":"10.1112/blms.70027","DOIUrl":"https://doi.org/10.1112/blms.70027","url":null,"abstract":"<p>The Algebraic Kirchberg–Phillips Question for Leavitt path algebras asks whether pointed <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-theory is a complete isomorphism invariant for unital, simple, purely infinite Leavitt path algebras over finite graphs. Most work on this problem has focused on determining whether (up to isomorphism) there is a unique unital, simple, Leavitt path algebra with trivial <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-theory (often reformulated as the question of whether the Leavitt path algebras <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$L_2$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <msub>\u0000 <mn>2</mn>\u0000 <mo>−</mo>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$L_{2_-}$</annotation>\u0000 </semantics></math> are isomorphic). However, it is unknown whether a positive answer to this special case implies a positive answer to the Algebraic Kirchberg–Phillips Question. In this note, we pose a different question that asks whether two particular non-simple Leavitt path algebras <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mo>∗</mo>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$L_k(mathbf {F}_*)$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mrow>\u0000 <mo>∗</mo>\u0000 <mo>∗</mo>\u0000 </mrow>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$L_k(mathbf {F}_{**})$</annotation>\u0000 </semantics></math> are isomorphic, and we prove that a positive answer to this question implies a positive answer to the Algebraic Kirchberg–Phillips Question.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1229-1248"},"PeriodicalIF":0.8,"publicationDate":"2025-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143836116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the minimal period of integer tilings","authors":"Izabella Łaba,&nbsp;Dmitrii Zakharov","doi":"10.1112/blms.70023","DOIUrl":"https://doi.org/10.1112/blms.70023","url":null,"abstract":"&lt;p&gt;If a finite set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$A$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; tiles the integers by translations, it also admits a tiling whose period &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has the same prime factors as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$|A|$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We prove that the minimal period of such a tiling is bounded by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;exp&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;log&lt;/mi&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;log&lt;/mi&gt;\u0000 &lt;mi&gt;log&lt;/mi&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$exp (c(log D)^2/log log D)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;annotation&gt;$D$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the diameter of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$A$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In the converse direction, given &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$epsilon &gt;0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, we construct tilings whose minimal period has the same prime factors as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$|A|$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and is bounded from below by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$D^{","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1160-1170"},"PeriodicalIF":0.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}