Advances in Mathematics最新文献

{"title":"Colored sl(N) homology and SU(N) representations","authors":"Joshua Wang","doi":"10.1016/j.aim.2025.110314","DOIUrl":"10.1016/j.aim.2025.110314","url":null,"abstract":"<div><div>We provide the first complete computations of colored <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math></span> homology for a nontrivial knot. In doing so, we show that the colored <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math></span> homology of the trefoil labeled by an exterior power of the defining representation is isomorphic to the cohomology of a closed manifold naturally associated to the trefoil. This manifold is the set of homomorphisms from the fundamental group of the complement of the trefoil to <span><math><mi>SU</mi><mo>(</mo><mi>N</mi><mo>)</mo></math></span> that send meridians to a particular conjugacy class depending on the label. We also provide complete computations and analogous isomorphisms for the first nontrivial link, the Hopf link.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110314"},"PeriodicalIF":1.5,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923442","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}

{"title":"Erratum to “Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds” [Adv. Math. 368 (2020) 107134]","authors":"Yalong Cao ,&nbsp;Jacob Gross ,&nbsp;Dominic Joyce","doi":"10.1016/j.aim.2025.110329","DOIUrl":"10.1016/j.aim.2025.110329","url":null,"abstract":"","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110329"},"PeriodicalIF":1.5,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922359","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}

{"title":"The coarse Baum-Connes conjecture with filtered coefficients and product metric spaces","authors":"Jianguo Zhang","doi":"10.1016/j.aim.2025.110327","DOIUrl":"10.1016/j.aim.2025.110327","url":null,"abstract":"<div><div>Inspired by the quantitative <em>K</em>-theory, in this paper, we introduce the coarse Baum-Connes conjecture with filtered coefficients which generalizes the original conjecture. There are two advantages for the conjecture with filtered coefficients. Firstly, the routes toward the coarse Baum-Connes conjecture also work for the conjecture with filtered coefficients. Secondly, the class of metric spaces that satisfy the conjecture with filtered coefficients is closed under products and yet it is unknown for the original conjecture. As an application, we discover some new examples of product metric spaces for the coarse Baum-Connes conjecture.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"475 ","pages":"Article 110327"},"PeriodicalIF":1.5,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922362","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}

{"title":"Asymptotic quantization of measures on Riemannian manifolds via covering growth estimates","authors":"Ata Deniz Aydın,&nbsp;Mikaela Iacobelli","doi":"10.1016/j.aim.2025.110311","DOIUrl":"10.1016/j.aim.2025.110311","url":null,"abstract":"<div><div>The quantization problem looks for best approximations of a probability measure on a given metric space by finitely many points, where the approximation error is measured with respect to the Wasserstein distance. On particular smooth domains, such as <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> or complete Riemannian manifolds, the quantization error is known to decay polynomially as the number of points is taken to infinity, provided the measure satisfies an integral condition which controls the amount of mass outside compact sets. On Riemannian manifolds, the existing integral condition involves a quantity measuring the growth of the exponential map, for which the only available estimates are in terms of lower bounds on sectional curvature.</div><div>In this paper, we provide a more general integral condition for the asymptotics of the quantization error on Riemannian manifolds, given in terms of the growth of the covering numbers of spheres, which is purely metric in nature and concerns only the large-scale growth of the manifold. We further estimate the covering growth of manifolds in two particular cases, namely lower bounds on the Ricci curvature and geometric group actions by a discrete group of isometries.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110311"},"PeriodicalIF":1.5,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917330","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}

{"title":"Relating ample and biample topological categories with Boolean restriction and range semigroups","authors":"Ganna Kudryavtseva","doi":"10.1016/j.aim.2025.110313","DOIUrl":"10.1016/j.aim.2025.110313","url":null,"abstract":"<div><div>We extend the equivalence by Cockett and Garner between restriction monoids and ample categories to the setting of Boolean range semigroups which are non-unital one-object versions of range categories. We show that Boolean range semigroups are equivalent to ample topological categories where the range map <em>r</em> is open, and étale Boolean range semigroups are equivalent to biample topological categories. These results yield the equivalence between étale Boolean range semigroups and Boolean birestriction semigroups and a characterization of when a Boolean restriction semigroup admits a compatible cosupport operation. We also recover the equivalence between Boolean birestriction semigroups and biample topological categories by Kudryavtseva and Lawson. Our technique builds on the usual constructions relating inverse semigroups with ample topological groupoids via germs and slices.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110313"},"PeriodicalIF":1.5,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143912236","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}

{"title":"Pentagonal number recurrence relations for p(n)","authors":"Kevin Gomez ,&nbsp;Ken Ono ,&nbsp;Hasan Saad ,&nbsp;Ajit Singh","doi":"10.1016/j.aim.2025.110308","DOIUrl":"10.1016/j.aim.2025.110308","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We revisit Euler's partition function recurrence, which asserts, for integers &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; is the &lt;em&gt;m&lt;/em&gt;th pentagonal number. We prove that this classical result is the &lt;span&gt;&lt;math&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; case of an infinite family of “pentagonal number” recurrences. For each &lt;span&gt;&lt;math&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we prove for positive &lt;em&gt;n&lt;/em&gt; that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Tr&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a divisor function, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Tr&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the &lt;em&gt;n&lt;/em&gt;th weight 2&lt;em&gt;ν&lt;/em&gt; Hecke trace of values of special twisted quadratic Dirichlet series, and each &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/m","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110308"},"PeriodicalIF":1.5,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907565","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}

{"title":"Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups","authors":"Jiayin Pan","doi":"10.1016/j.aim.2025.110310","DOIUrl":"10.1016/j.aim.2025.110310","url":null,"abstract":"<div><div>We study the fundamental group of an open <em>n</em>-manifold <em>M</em> of nonnegative Ricci curvature with additional stability conditions on <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>, the Riemannian universal cover of <em>M</em>. We prove that if every asymptotic cone of <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric cone, then <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is finitely generated and contains a normal abelian subgroup of finite index; if in addition <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> has Euclidean volume growth of constant at least <em>L</em>, then we can bound the index of that abelian subgroup by a constant <span><math><mi>C</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>L</mi><mo>)</mo></math></span>. In particular, our result implies that if <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> has Euclidean volume growth of constant at least <span><math><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, then <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is finitely generated and <span><math><mi>C</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>-abelian.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110310"},"PeriodicalIF":1.5,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907563","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}

{"title":"Lp maximal bounds and Sobolev regularity of two-parameter averages over tori","authors":"Juyoung Lee ,&nbsp;Sanghyuk Lee","doi":"10.1016/j.aim.2025.110312","DOIUrl":"10.1016/j.aim.2025.110312","url":null,"abstract":"<div><div>We investigate <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> boundedness of the maximal function defined by the averaging operator <span><math><mi>f</mi><mo>↦</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mi>f</mi></math></span> over the two-parameter family of tori<span><span><span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>:</mo><mo>=</mo><mo>{</mo><mo>(</mo><mo>(</mo><mi>t</mi><mo>+</mo><mi>s</mi><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>)</mo><mi>cos</mi><mo>⁡</mo><mi>ϕ</mi><mo>,</mo><mspace></mspace><mo>(</mo><mi>t</mi><mo>+</mo><mi>s</mi><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>)</mo><mi>sin</mi><mo>⁡</mo><mi>ϕ</mi><mo>,</mo><mspace></mspace><mi>s</mi><mi>sin</mi><mo>⁡</mo><mi>θ</mi><mo>)</mo><mo>:</mo><mi>θ</mi><mo>,</mo><mi>ϕ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo>)</mo><mo>}</mo></math></span></span></span> with <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>t</mi><mo>&gt;</mo><mi>s</mi><mo>&gt;</mo><mn>0</mn></math></span> for some <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We prove that the associated (two-parameter) maximal function is bounded on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> if and only if <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span>. Also, we obtain <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>–<span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span> bounds for the local maximal operator on a sharp range of <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span>. Furthermore, sharp smoothing estimates are obtained including the local smoothing estimates for the operators <span><math><mi>f</mi><mo>↦</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mi>f</mi></math></span> and <span><math><mi>f</mi><mo>↦</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>t</mi></mrow></msubsup><mi>f</mi></math></span>. For these purposes, we make use of Bourgain–Demeter's decoupling inequality and Guth–Wang–Zhang's local smoothing estimate for the 2-dimensional wave operator.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110312"},"PeriodicalIF":1.5,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907566","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}

{"title":"Period function of Maass forms from Ramanujan's lost notebook","authors":"YoungJu Choie ,&nbsp;Rahul Kumar","doi":"10.1016/j.aim.2025.110317","DOIUrl":"10.1016/j.aim.2025.110317","url":null,"abstract":"<div><div>The Lost Notebook of Ramanujan contains a number of beautiful formulas, one of which can be found on its page 220. It involves an interesting function, which we denote as <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. In this paper, we show that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> belongs to the category of period functions as it satisfies the period relations of Maass forms in the sense of Lewis and Zagier <span><span>[11]</span></span>. Hence, we refer to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> as the <em>Ramanujan period function</em>. Moreover, one of the salient aspects of the Ramanujan period function <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> that we found out is that it is a Hecke eigenfunction under the action of Hecke operators on the space of periods. We also establish that it naturally appears in a Kronecker limit formula of a certain zeta function, revealing its connections to various topics. Finally, we generalize <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> to include a parameter <em>s</em>, connecting our work to the broader theory of period functions developed by Bettin and Conrey <span><span>[4]</span></span> and Lewis and Zagier <span><span>[11]</span></span>. We emphasize that Ramanujan was the first to study this function, marking the beginning of the study of period functions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110317"},"PeriodicalIF":1.5,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907567","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}

{"title":"A dimensional mass transference principle for Borel probability measures and applications","authors":"Edouard Daviaud","doi":"10.1016/j.aim.2025.110304","DOIUrl":"10.1016/j.aim.2025.110304","url":null,"abstract":"<div><div>In this article, we establish a dimensional mass transference principle valid when the ambient measure is finite. We provide two applications of this result. First we study certain dynamical coverings associated with some self-similar IFS with overlaps and then we give an application in Diophantine approximation to rational approximation among points of <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> with few digit equal to 1 in their base-3 expansion.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"474 ","pages":"Article 110304"},"PeriodicalIF":1.5,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907564","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}