We study Rk×Zℓmathbb {R}^k times mathbb {Z}^ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program.
我们研究了rk × zr mathbb {R} k 乘以mathbb {Z}^的作用在任意紧流形上的投影密集的Anosov元素集和一维粗糙Lyapunov叶。这样的行为被称为完全的Cartan行为。我们将这类动作完全分类为低维Anosov流、微分同态和仿射动作，验证了该类的Katok-Spatzier猜想。这是通过引入一个新工具来实现的，即动态定义的拓扑群的作用，它描述了粗糙Lyapunov叶中的路径，并理解了它的生成器和关系。我们获得了季默程序的应用程序。
{"title":"Cartan actions of higher rank abelian groups and their classification","authors":"Ralf Spatzier, Kurt Vinhage","doi":"10.1090/jams/1033","DOIUrl":"https://doi.org/10.1090/jams/1033","url":null,"abstract":"We study <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript k Baseline times double-struck upper Z Superscript script l\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mi>ℓ<!-- ℓ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {R}^k times mathbb {Z}^ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135782996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is such a manifold, we show that the type D structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper C upper F upper D With caret left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>F</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">widehat {CFD}(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may be viewed as a set of immersed curves decorated with local systems in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper M"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">partial M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H upper F With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>F</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">widehat {HF}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decreases under a certain class of degree one map
{"title":"Bordered Floer homology for manifolds with torus boundary via immersed curves","authors":"Jonathan Hanselman, Jacob Rasmussen, Liam Watson","doi":"10.1090/jams/1029","DOIUrl":"https://doi.org/10.1090/jams/1029","url":null,"abstract":"This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is such a manifold, we show that the type D structure <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper C upper F upper D With caret left-parenthesis upper M right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mover> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>F</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">widehat {CFD}(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may be viewed as a set of immersed curves decorated with local systems in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential upper M\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">partial M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h\"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=\"application/x-tex\">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h\"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=\"application/x-tex\">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper H upper F With caret\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mover> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>F</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">widehat {HF}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decreases under a certain class of degree one map","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135520700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the geometric wave-front set of specific half-integral-depth supercuspidal representations of ramified pp-adic unitary groups is not a singleton.
我们证明了分支p进酉群的特定半积分深度超尖表示的几何波前集不是单态的。
{"title":"Geometric wave-front set may not be a singleton","authors":"Cheng-Chiang Tsai","doi":"10.1090/jams/1031","DOIUrl":"https://doi.org/10.1090/jams/1031","url":null,"abstract":"We show that the geometric wave-front set of specific half-integral-depth supercuspidal representations of ramified <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic unitary groups is not a singleton.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135063301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bryna Kra, Joel Moreira, Florian Richter, Donald Robertson
Motivated by questions asked by Erdős, we prove that any set A⊂NAsubset mathbb {N} with positive upper density contains, for any k∈Nkin mathbb {N}, a sumset B1+⋯+BkB_1+cdots +B_k, where B1B_1, …, Bk⊂NB_ksubset mathbb {N} are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of k=2k=2.
受Erdős提出的问题的启发，我们证明了对于任意k∈N kin mathbb {N}，具有正上密度的任何集合A∧A子集mathbb {N}包含一个sumset b1 +⋯+B k B_1+cdots +B_k，其中b1 B_1，…，B k∧N B_k子集mathbb {N}是无限的。我们的证明使用遍历理论并依赖于测度保持系统的结构结果。我们的技术是新的，即使对于以前已知的k=2 k=2的情况。
{"title":"Infinite sumsets in sets with positive density","authors":"Bryna Kra, Joel Moreira, Florian Richter, Donald Robertson","doi":"10.1090/jams/1030","DOIUrl":"https://doi.org/10.1090/jams/1030","url":null,"abstract":"Motivated by questions asked by Erdős, we prove that any set <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A subset-of double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Asubset mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with positive upper density contains, for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k element-of double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">kin mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a sumset <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B 1 plus midline-horizontal-ellipsis plus upper B Subscript k\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B_1+cdots +B_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B 1\"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">B_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Subscript k Baseline subset-of double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B_ksubset mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k equals 2\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135396765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the natural analogue of the classical Kakeya conjecture over the pp-adic numbers.
在p进数上证明了经典Kakeya猜想的自然类比。
{"title":"The 𝑝-adic Kakeya conjecture","authors":"Bodan Arsovski","doi":"10.1090/jams/1021","DOIUrl":"https://doi.org/10.1090/jams/1021","url":null,"abstract":"We prove the natural analogue of the classical Kakeya conjecture over the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic numbers.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"199 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135813111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a “soft” proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg’s later proofs.
{"title":"On the meromorphic continuation of Eisenstein series","authors":"Joseph Bernstein, Erez Lapid","doi":"10.1090/jams/1020","DOIUrl":"https://doi.org/10.1090/jams/1020","url":null,"abstract":"Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a “soft” proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg’s later proofs.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136086654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {C}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of cycle lengths in a graph <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Subscript normal o normal d normal d Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {C}_{mathrm {odd}}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of odd numbers in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {C}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has chromatic number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-
1981年，Erdős和Hajnal提出了一个问题:在一个色数为无穷大的图中，奇循环长度的倒数之和是否必然是无穷大的?设C(G) mathcal C{(G)是图G G中循环长度的集合，设C odd(G) }mathcal C＿{}{mathrm odd{(G)是C(G)中奇数的集合}}mathcal C{(G)。我们证明了，如果G G有色数k k，则∑r∈C odd(G)1/ r≥(1/2−o k(1)) log (k }sum ＿ {ellinmathcal C＿{}{mathrm odd{(G)}}1/}ellgeq (1/2-o k(1)) log k。这解决了Erdős和Hajnal的奇循环问题，并且，更进一步，这个界是渐近最优的。1984年，Erdős问是否存在这样的d d，即每个色数至少为d d(或者甚至可能只有平均度至少为d d)的图都有一个周期，其长度是2的幂。我们证明了平均次条件对于这个问题是充分的，并且用除2的幂之外的适用于广泛序列的方法来解决它。最后，我们用我们的方法来证明，对于每k k，存在一些d d，使得每一个平均度至少为d d的图都有一个完整图k k k k k k k的细分，其中每条边被细分的次数相同。这证实了托马森1984年的一个猜想。
{"title":"A solution to Erdős and Hajnal’s odd cycle problem","authors":"Hong Liu, Richard Montgomery","doi":"10.1090/jams/1018","DOIUrl":"https://doi.org/10.1090/jams/1018","url":null,"abstract":"In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis upper G right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {C}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of cycle lengths in a graph <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C Subscript normal o normal d normal d Baseline left-parenthesis upper G right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">o</mml:mi> <mml:mi mathvariant=\"normal\">d</mml:mi> <mml:mi mathvariant=\"normal\">d</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {C}_{mathrm {odd}}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of odd numbers in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis upper G right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {C}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that, if <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has chromatic number <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135822249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：